Optimized adaptive asymptotic control for leaderless multi-agent systems under deception attacks

Zhixu Du; Hao Zhang; Zhuping Wang; Sheng Gao

doi:10.1051/sands/2025013

Security and Safety for Next Generation Industrial Systems

Open Access

Issue		Security and Safety Volume 4, 2025 Security and Safety for Next Generation Industrial Systems


Article Number		2025013
Number of page(s)		21
Section		Industrial Control
DOI		https://doi.org/10.1051/sands/2025013
Published online		27 October 2025

Security and Safety, Vol. 4, 2025013 (2025)

Research Article

Optimized adaptive asymptotic control for leaderless multi-agent systems under deception attacks

Zhixu Du¹, Hao Zhang¹^,2^*, Zhuping Wang¹^,2 and Sheng Gao³

¹ Shanghai Research Institute for Intelligent Autonomous Systems, Tongji University, Shanghai, 200092, China
² College of Electronic and Information Engineering, the Department of Control Science and Engineering, Tongji University, Shanghai, 200092, China
³ School of Information Science and Engineering, East China University of Science and Technology, Shanghai, 200237, China

^* Corresponding author (email: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 27 March 2025
Revised: 19 July 2025
Accepted: 15 September 2025

Abstract

This article concentrate on optimized adaptive consensus control problem for the nonlinear multi-agent systems subjected to deception attacks. To mitigate the influence of false data on the system, the transformed tracking error is formulated under the backstepping technique, and the optimized consensus control strategy incorporating Nussbaum technique is utilized to eliminate the destabilizing effects of time-varying gains in the attack. The proposed critic-actor structured reinforcement learning optimal algorithm executes control actions via the actor neural network and evaluates system performance through the critic neural network, enabling the closed-loop system to achieve optimized tracking control performance. Based on the Lyapunov stability method, it is demonstrated that all signals bounded within the closed-loop system, ensuring the achievement of asymptotic output consensus. To illustrate the efficacy of the proposed control approach, some simulation results are finally presented.

Key words: Cooperative control / Critic-actor architecture / Deception attacks / Optimized backstepping technique

Citation: Du Z, Zhang H and Wang Z et al. Optimized adaptive asymptotic control for leaderless multi-agent systems under deception attacks. Security and Safety 2025; 4: 2025013. https://doi.org/10.1051/sands/2025013

© The Author(s) 2025. Published by EDP Sciences and China Science Publishing & Media Ltd.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Due to its extensive potential in real-world applications, the field of cooperative control in multi-agent systems (MASs) has recently garnered widespread notice [1–6]. As a cornerstone challenge in cooperative control, the consensus control strategy forces a cluster of agents to reach a common state through appropriate control strategies. The consensus challenges in MASs have been thoroughly investigated over the years [7–10]. For example, in [11], the authors present a power-efficient secure full-duplex integrated sensing and communication framework that ensures uplink and downlink secrecy while enabling radar sensing of eavesdroppers through joint beamforming and artificial noise design. The adaptive neural-network-based distributed control approach was introduced for the nonlinear MASs with unmodeled dynamics in [12]. Liang et al. [13] introduced the adaptive consensus control method of the nonaffine MASs subject to dynamic disturbance. However, the above achievements in [8–10, 12, 13] cannot obtain the optimal solution. Therefore, it is crucial to discover an appropriate control technique for MASs that can achieve the control goal while minimizing energy usage.

Optimal control, which can get the optimal solution by consuming minimal cost, has attracted significant attention. Theoretically, Hamilton-Jacobi-Bellman (HJB) equation is a powerful tool to handle the optimal control problems [14–16]. Nonetheless, it is challenging to derive an analytical solution because of the high degree of nonlinearity and intractability. Consequently, policy iteration and reinforcement learning (RL) pave the way to address such challenging problem. The RL algorithm is a potent tool in dealing with the adaptive optimal control problems based on actor-critic architectures. The HJB equation for optimal control problem was solved by actor-critic-identifier architecture in [17], where the identifier radial basis function neural network (RBF NN) was employed to approximate the unknown nonlinearities term. Wen et al. [18] examined the optimal control problem for a single system by employing the simplified RL based on actor-critic-identifier architecture. The adaptive control problem of discrete-time MASs suffering from actuator faults was addressed in [19] by utilizing a reinforcement learning strategy. In [20], the integration of a boundary function with a neural network yielded a robust cost control strategy for uncertain systems.

In spite of the progress, the results mentioned above did not consider the question of attacks that need to be considered in cyber-physical systems [21–26]. The deception attacks means that attackers maliciously transmitting false sensor data to the controller when sensors are attacked, potentially leading to a decline in control performance and even to system instability. To solve the difficulty, a number of useful methods have been proposed. A detection method was proposed to detect deception attacks for the cyber-physical systems in [27]. A recursive filtering approach was introduced in [28] to handle stochastic systems compromised by the cyber attacks. In the case of the attackers send false sensor data, Ren et al. [29] formulated an adaptive control algorithm by utilizing the boundedness property of the Nussbaum function, which resisted the potential impact of external attacks on the stability of the controller. Taking the presence of deception attacks into account for uncertain nonlinear systems, an anti-attack controller was studied in [30] using a backstepping technique. Due to the uncertainty of state feedback coefficients altered by deception attacks, there are limited mathematical tools available to optimally guarantee the asymptotic stability of the MASs.

Motivated by this, this article is centered around the optimized adaptive cooperative control for the nonlinear MASs subject to deception attacks. In contrast to [28], the recommended control scheme can realize asymptotic output consensus by introducing the Nussbaum function and constructing a special synchronization error, which effectively removes the restrictive condition that the state feedback coefficients caused by deception attacks in [28] are compelled to be known. One more thing, by constructing a special synchronization error, the effect of the false state information can be effectively eliminated, leading to asymptotic consensus among the outputs of all subsystems, which cannot be done by the existing control strategies in [28–30]. Additionally, by adopting the negative gradient of a positive function to design the update rate, which effectively circumvents the computational burden associated with utilizing the gradient descent algorithm.

The organization of this article is outlined: Section 2 covers the preliminary concepts; the adaptive optimal controller is constructed by the action of Nussbaum technique and NN-based RL in Section 3 and a illustrative example based investigation is carried out in Section 4; the overall content is summarized in Section 5.

2. Preliminaries

2.1. Algebraic graph theory

We consider a topology structure composed of N agents with strong connectivity properties to each other. The topology of this group is described by 𝒢 = (𝒵,ℒ,𝒜), where ℒ ∈ 𝒵 × 𝒵 indicates the edge set, and 𝒵 = {𝒵_i|i∈I[1,N]} denotes the node set. We use (𝒵_j,𝒵_i) ∈ ℒ to represent a directed link, which implies that agent i transmits the message directly to agent j. Let N_i = {𝒵_j ∣ (𝒵_j, 𝒵_i ∈ ℒ, i ≠ j)} represents all the neighbor nodes of the i-th agent. A adjacency matrix 𝒜 = [a_i, j] ∈ R^N × N is associated to graph 𝒢 with elements a_i, j > 0, else a_i, j = 0. Clearly, the diagonal elements a_i, i = 0. The diagonal matrix is represented as 𝒟 = diag{d₁, …, d_N}, where $d_{i} = \sum_{j = 1}^{N} a_{i, j}$ $Mathematical equation: $ d_{i}=\sum_{j=1}^{N}a_{i,j} $$ . The Laplacian matrix is ℘ = 𝒟 − 𝒜.

2.2. Problem statement

Consider the following leaderless nonlinear MASs with deception attacks:

$\begin{matrix} \begin{matrix} {\begin{matrix} {\dot{ζ}}_{i, l} & = ζ_{i, l + 1} + g_{i, l} ({\bar{ζ}}_{i, l}), \\ {\dot{ζ}}_{i, m} & = u_{i} + g_{i, m} ({\bar{ζ}}_{i, m}), \\ y_{i} & = ζ_{i, 1}, i = 1, 2, \dots, N, l = 1, 2, \dots, m - 1, \end{matrix} \end{matrix} \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \begin{split} \left\{ \begin{aligned} \dot{\zeta }_{i,l}&= \zeta _{i,l+1}+g_{i,l}({\bar{\zeta }_{i,l}}), \\ \dot{\zeta }_{i,m}&= u_{i}+g_{i,m}({\bar{\zeta }_{i,m}}),\\ y_{i}&= \zeta _{i,1} , i=1,2, \ldots ,N,\quad l=1,2,\ldots , m-1, \end{aligned} \right. \end{split} \end{aligned} $$$ (1)

where ${\bar{ζ}}_{i, k} = {[ζ_{i, 1}, η_{i, 2}, \dots, ζ_{i, k}]}^{T} \in R^{k} (k = 1, 2, \dots, m)$ $Mathematical equation: $ {\bar \zeta_{i,k}}=[\zeta_{i,1},\eta_{i,2},\ldots,\zeta_{i,k}]^{T}\in {R^k} (k=1,2,\ldots,m) $$ denote the state vectors, $g_{i, k} ({\bar{ζ}}_{i, k}) : R^{m} \to R$ $Mathematical equation: $ g_{i,k}({\bar \zeta_{i,k}}):{R^m} \to R $$ depicts continuous functions with nonlinear properties, u_i ∈ R is the input, y_i ∈ R is the output. When the sensor is attacked, the tampered measurement ${\hat{ζ}}_{i, k} (t)$ $Mathematical equation: $ \hat{\zeta}_{i,k}(t) $$ is the only signal that can be received. The available compromised system states ${\hat{ζ}}_{i, k} (t)$ $Mathematical equation: $ \hat{\zeta}_{i,k}(t) $$ can be written as

$\begin{matrix} {\hat{ζ}}_{i, k} (t) = ζ_{i, k} (t) + ψ (ζ_{i, k} (t), t) = (1 + {\bar{ν}}_{i, k} (t)) ζ_{i, k} (t), \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \hat{\zeta }_{i,k}(t)=\zeta _{i,k}(t)+\psi \big (\zeta _{i,k}(t),t\big ) =\big (1+\bar{\nu }_{i,k}(t)\big )\zeta _{i,k}(t), \end{aligned} $$$

where ψ(ζ_i, k(t),t) capture sensor attacks, $ψ (ζ_{i, k} (t), t) = {\bar{ν}}_{i, k} (t) ζ_{i, k} (t)$ $Mathematical equation: $ \psi(\zeta_{i,k}(t),t)=\bar{\nu}_{i,k}(t)\zeta_{i,k}(t) $$ , and ${\bar{ν}}_{i, k} (t)$ $Mathematical equation: $ \bar{\nu}_{i,k}(t) $$ is a time varying bounded variable. Then, we have

$\begin{matrix} ζ_{i, k} = ϖ_{i, k} {\hat{ζ}}_{i, k}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \zeta _{i,k}=\varpi _{i,k} \hat{\zeta }_{i,k}, \end{aligned} $$$

where $ϖ_{i, k} = \frac{1}{1 + {\bar{ν}}_{i, k} (t)}$ $Mathematical equation: $ \varpi_{i,k}=\frac{1}{1+\bar{\nu}_{i,k}(t)} $$ . Thus, the leaderless nonlinear MASs under deception attacks becomes

$\begin{matrix} \begin{matrix} {\begin{matrix} {\dot{\hat{ζ}}}_{i, l} & = {\hat{ζ}}_{i, l + 1} + \frac{g_{i, l}}{ϖ_{i, l}} - \frac{{\dot{ϖ}}_{i, l} {\hat{ζ}}_{i, l}}{ϖ_{i, l}}, \\ {\dot{\hat{ζ}}}_{i, m} & = \frac{u_{i}}{ϖ_{i, m}} + \frac{g_{i, m}}{ϖ_{i, m}} - \frac{{\dot{ϖ}}_{i, m} {\hat{ζ}}_{i, m}}{ϖ_{i, m}} . \end{matrix} \end{matrix} \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \begin{split} \left\{ \begin{aligned} \dot{\hat{\zeta }}_{i,l}&=\hat{\zeta }_{i,l+1}+\frac{g_{i,l}}{\varpi _{i,l}}-\frac{\dot{\varpi }_{i,l}\hat{\zeta }_{i,l}}{\varpi _{i,l}}, \\ \dot{\hat{\zeta }}_{i,m}&=\frac{u_{i}}{\varpi _{i,m}} +\frac{g_{i,m}}{\varpi _{i,m}}-\frac{\dot{\varpi }_{i,m}\hat{\zeta }_{i,m}}{\varpi _{i,m}}. \end{aligned} \right. \end{split} \end{aligned} $$$ (2)

To accomplish these objectives, we rely on the following two standard assumptions and one lemma regarding system dynamics.

Assumption 1 [12]: The leader’s output is m-order continuous differentiable.

Assumption 2 [29, 30]: The variable ${\bar{ν}}_{i, k} (t)$ $Mathematical equation: $ \bar{\nu}_{i,k}(t) $$ satisfies ${\bar{ν}}_{i, k} (t) \neq - 1$ $Mathematical equation: $ \bar{\nu}_{i,k}(t)\neq-1 $$ . |ϖ_{_i, k}(t)| ≤ ϖ_max and ${\dot{ϖ}}_{_{i, k}} (t)$ $Mathematical equation: $ \dot{\varpi}_{_{i,k}}(t) $$ are bounded with ϖ_max being unknown constants.

This Assumption 2 ensures that $1 + {\bar{ν}}_{i, k} (t) \neq 0$ $Mathematical equation: $ 1 + \bar{\nu}_{i,k}(t) \neq 0 $$ , preventing a singularity in which the attack gain ${\bar{ν}}_{i, k} (t) = - 1$ $Mathematical equation: $ \bar{\nu}_{i,k}(t) = -1 $$ would cancel the system state ( ${\hat{ζ}}_{i, k} = 0$ $Mathematical equation: $ \hat{\zeta}_{i,k} = 0 $$ ), resulting in a complete loss of state information.

Definition 1

[32] If N(τ) is a Nussbaum-type function, such that the following formulas hold

$\begin{matrix} {\begin{matrix} lim_{a \to \infty} sup \frac{1}{a} \int_{0}^{a} N (τ) d τ = + \infty, \\ lim_{a \to \infty} inf \frac{1}{a} \int_{0}^{a} N (τ) d τ = - \infty, \end{matrix} \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \left\{ \begin{aligned} \lim _{a\rightarrow \infty }\sup \frac{1}{a}\int ^{a}_{0}N(\tau )d\tau =+\infty ,\\ \lim _{a\rightarrow \infty }\inf \frac{1}{a}\int ^{a}_{0}N(\tau )d\tau =-\infty , \end{aligned} \right. \end{aligned} $$$ (3)

and we select N(τ)=expτ²sin(π τ) in the paper.

Lemma 1

[33] Define smooth function V(t)≥0 and ν(t) on [0, t_a] with their initials τ(0) being bounded. For ${\bar{ς}}_{a} > 0$ $Mathematical equation: $ \bar{\varsigma}_{a} > 0 $$ and ${\bar{ς}}_{b} > 0$ $Mathematical equation: $ \bar{\varsigma}_{b} > 0 $$ , if exists a Nussbaum function N(ν(t)) satisfies

$\begin{matrix} V (t) \leq {\bar{ς}}_{a} + \int_{0}^{t} [ν (\bar{κ}) N (ν) + {\bar{ς}}_{b}] \dot{ν} d \bar{κ}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} V(t)\le \bar{\varsigma }_{a}+\int ^{t}_{0}[\nu (\bar{\kappa }) N(\nu )+\bar{\varsigma }_{b}]\dot{\nu }d\bar{\kappa }, \end{aligned} $$$

where $ν (\bar{κ}) > 0$ $Mathematical equation: $ \nu(\bar{\kappa}) > 0 $$ is a bounded varying function, then V(t), ν(t) and $\int_{0}^{t} [ν (\bar{κ}) N (ν) + {\bar{ς}}_{b}] \dot{ν} d \bar{κ}$ $Mathematical equation: $ \int^{t}_{0}[\nu(\bar{\kappa}) N(\nu)+\bar{\varsigma}_{b}]\dot{\nu}d\bar{\kappa} $$ must be bounded on [0, t_a].

Before the attack compensation optimization controller is designed, some relevant constants some constants are proposed to facilitate arithmetic. ${\bar{\hat{ζ}}}_{i, k} = {[{\hat{ζ}}_{i, 1}, {\hat{ζ}}_{i, 2}, \dots, {\hat{ζ}}_{i, k}]}^{T}$ $Mathematical equation: $ \bar{\hat{\zeta}}_{i,k}=[\hat{\zeta}_{i,1}, \hat{\zeta}_{i,2}, \ldots, \hat{\zeta}_{i,k}]^{T} $$ represent the available compromised system states vectors, where k = 1, 2, …, m. ϑ_i, k and ϑ_{i, m + 1} are defined as $ϑ_{i, k} = ϖ_{max} | | ξ_{i, k}^{*} | |, ϑ_{i, m + 1} = ϖ_{max}^{2}$ $Mathematical equation: $ \vartheta_{i,k}=\varpi_{\max}\vert\vert\xi^{*}_{i,k}\vert\vert, \quad \vartheta_{i,m+1}=\varpi_{\max}^{2} $$ , where $ξ_{i, k}^{*}$ $Mathematical equation: $ \xi^{*}_{i,k} $$ are the ideal weight vectors. We define ${\hat{ϑ}}_{i, k}$ $Mathematical equation: $ \hat{\vartheta}_{i,k} $$ is the estimation of θ_i, k, and ${\tilde{ϑ}}_{i, k}$ $Mathematical equation: $ \tilde{\vartheta}_{i,k} $$ is the approximation error with ${\tilde{ϑ}}_{i, k} = {\hat{ϑ}}_{i, k} - ϑ_{i, k}$ $Mathematical equation: $ \tilde{\vartheta}_{i,k}=\hat{\vartheta}_{i,k}-\vartheta_{i,k} $$ . We define that ${\hat{ϑ}}_{i, m + 1}$ $Mathematical equation: $ \hat{\vartheta}_{i,m+1} $$ represent the estimation of ϑ_{i, m + 1}, and ${\tilde{ϑ}}_{i, m + 1}$ $Mathematical equation: $ \tilde{\vartheta}_{i,m+1} $$ is the approximation error with ${\tilde{ϑ}}_{i, m + 1} = {\hat{ϑ}}_{i, m + 1} - ϑ_{i, m + 1}$ $Mathematical equation: $ \tilde{\vartheta}_{i,m+1}=\hat{\vartheta}_{i,m+1}-\vartheta_{i,m+1} $$ . $W_{J i, k}^{*} \in R^{q_{1}}$ $Mathematical equation: $ W^{*}_{Ji,k}\in R^{q_{1}} $$ is the ideal weight vector and ${\bar{R}}_{J i, k} ({\bar{\hat{ζ}}}_{i, k}, {\hat{s}}_{i, k}) \in R^{q_{1}}$ $Mathematical equation: $ \bar{R}_{Ji,k}(\bar{\hat{\zeta}}_{i,k},\hat{s}_{i,k})\in R^{q_{1}} $$ depicts the basis function vector. Moreover, ρ_i, l and ρ_i, m are defined as $ρ_{i, l} = ϖ_{max} (ε_{i, l} + | \frac{1}{2} ϖ_{i, l} {\hat{W}}_{a i, l}^{T} (t) {\bar{R}}_{J i, l} |), ρ_{i, m} = ϖ_{max} (ε_{i, m} + | \frac{1}{2} {\hat{W}}_{a i, m}^{T} (t) {\bar{R}}_{J i, m} |)$ $Mathematical equation: $ \rho_{i,l}= \varpi_{\max}\Big(\varepsilon_{i,l} + \vert\frac{1}{2}\varpi_{i,l}\hat{W}^{T}_{ai,l}(t)\bar{R}_{Ji,l}\vert\Big), \rho_{i,m}=\varpi_{\max}\Big(\varepsilon_{i,m} + \vert\frac{1}{2}\hat{W}^{T}_{ai,m}(t)\bar{R}_{Ji,m}\vert\Big) $$ , where l = 1, …, m − 1. Then, we define that ${\hat{ρ}}_{i, k} (k = 1, 2, \dots, m)$ $Mathematical equation: $ \hat{\rho}_{i,k}(k=1,2,\ldots,m) $$ are the estimations of ρ_i, k, and ${\tilde{ρ}}_{i, k}$ $Mathematical equation: $ \tilde{\rho}_{i,k} $$ are the errors with ${\tilde{ρ}}_{i, k} = {\hat{ρ}}_{i, k} - ρ_{i, k}$ $Mathematical equation: $ \tilde{\rho}_{i,k}=\hat{\rho}_{i,k}-\rho_{i,k} $$ .

Control objective: The goal is to design an optimal controller by minimizing the performance function associated with system (2), such that:

(1) The signals ${\tilde{ρ}}_{i, k}$ $Mathematical equation: $ \tilde{\rho}_{i,k} $$ , ${\tilde{W}}_{c i, k}$ $Mathematical equation: $ \tilde{W}_{ci,k} $$ , ${\tilde{W}}_{a i, k}$ $Mathematical equation: $ \tilde{W}_{ai,k} $$ , ${\hat{ϑ}}_{i, k}$ $Mathematical equation: $ \hat{\vartheta}_{i,k} $$ , ${\hat{ρ}}_{i, k}$ $Mathematical equation: $ \hat{\rho}_{i,k} $$ , ${\hat{ϑ}}_{i, m + 1}$ $Mathematical equation: $ \hat{\vartheta}_{i,m+1} $$ , ${\hat{W}}_{c i, k}$ $Mathematical equation: $ \hat{W}_{ci,k} $$ , ${\hat{W}}_{a i, k}$ $Mathematical equation: $ \hat{W}_{ai,k} $$ , ζ_i, k, and u_i are all bounded, while ${\hat{ζ}}_{i, 1} - {\hat{ζ}}_{j, 1}$ $Mathematical equation: $ \hat{\zeta}_{i,1}-\hat{\zeta}_{j,1} $$ , and ζ_i, 1 − ζ_j, 1 converge to zero as t → ∞.

(2) Asymptotic consensus among the outputs of all subsystems are achieved.

3. Main results

This section focuses on the progressive design of an optimized adaptive control method for the nonlinear leaderless MASs subjected to deceptive attacks. In the context of controller design for the i-th agent, cooperative errors and virtual tracking errors are respectively given as

$\begin{matrix} {\hat{s}}_{i, 1} & = {\hat{ζ}}_{i, 1} - {\hat{z}}_{i, 1}, {\dot{\hat{z}}}_{i, 1} = - \sum_{j = 1}^{N} a_{i, j} ({\hat{ζ}}_{i, 1} - {\hat{ζ}}_{j, 1}), \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \hat{s}_{i,1}&=\hat{\zeta }_{i,1}-\hat{z}_{i,1}, \dot{\hat{z}}_{i,1}=-\sum \nolimits _{j = 1}^N {{a_{i,j}}} (\hat{\zeta }_{i,1}-\hat{\zeta }_{j,1}),\end{aligned} $$$ (4)

$\begin{matrix} {\hat{s}}_{i, l} & = {\hat{ζ}}_{i, l} - α_{i, l - 1}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \hat{s}_{i,l}&=\hat{\zeta }_{i,l}-\alpha _{i,l-1}, \end{aligned} $$$ (5)

where ${\hat{z}}_{i, 1} (0) = {\hat{ζ}}_{i, 1} (0)$ $Mathematical equation: $ \hat{z}_{i,1}(0)=\hat{\zeta}_{i,1}(0) $$ , and α_{i, l − 1} (l = 2, 3, …, m), is the virtual controller. Based on $s_{i, k} = ϖ_{i, k} {\hat{s}}_{i, k} (k = 1, 2, \dots, m)$ $Mathematical equation: $ s_{i,k}=\varpi_{i,k}\hat{s}_{i,k} \quad (k=1,2,\ldots, m) $$ and (5), follows that

$\begin{matrix} \begin{matrix} {\begin{matrix} s_{i, 1} & = ζ_{i, 1} - ϖ_{i, 1} {\hat{z}}_{i, 1}, \\ s_{i, l} & = ζ_{i, l} - ϖ_{i, l} α_{i, l - 1} . \end{matrix} \end{matrix} \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \begin{split} \left\{ \begin{aligned} s_{i,1}&=\zeta _{i,1}-\varpi _{i,1} \hat{z}_{i,1}, \\ s_{i,l}&=\zeta _{i,l}-\varpi _{i,l} \alpha _{i,l-1}. \end{aligned} \right. \end{split} \end{aligned} $$$ (6)

Remark2 The attackers send false state information to the controller when the sensors are attacked. We reveal that the traditional backstepping approach requires real state information x_i, l in the synchronization error for MASs with l = 1, …, m, which cannot be directly applied. By constructing a special synchronization error and introducing the Nussbaum function technique, our proposed control scheme can achieve asymptotic output consensus.

Step 1: From (2) and (4), the following relation can be derived

$\begin{matrix} {\dot{\hat{s}}}_{i, 1} = {\hat{ζ}}_{i, 2} + \frac{g_{i, 1}}{ϖ_{i, 1}} - \frac{{\dot{ϖ}}_{i, 1} {\hat{ζ}}_{i, 1}}{ϖ_{i, 1}} + \sum_{j = 1}^{N} a_{i, j} ({\hat{ζ}}_{i, 1} - {\hat{ζ}}_{j, 1}) . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{\hat{s}}_{i,1}=\hat{\zeta }_{i,2}+\frac{g_{i,1}}{\varpi _{i,1}}-\frac{\dot{\varpi }_{i,1}\hat{\zeta }_{i,1}}{\varpi _{i,1}} +\sum ^{N}_{j=1}a_{i,j}\big (\hat{\zeta }_{i,1}-\hat{\zeta }_{j,1}\big ). \end{aligned} $$$

The performance index $J_{i, 1} ({\hat{s}}_{i, 1})$ $Mathematical equation: $ J_{i,1}(\hat{s}_{i,1}) $$ integrates the tracking error ${\dot{\hat{s}}}_{i, 1}$ $Mathematical equation: $ \dot{\hat{s}}_{i,1} $$ and the optimized virtual input $α_{i, 1}^{*}$ $Mathematical equation: $ \alpha^{*}_{i,1} $$ to balance tracking accuracy and control cost.

$\begin{matrix} J_{i, 1}^{*} ({\hat{s}}_{i, 1}) = min_{α_{i, 1} \in Ψ (Ω)} (\int_{t}^{\infty} K_{i, 1} ({\hat{s}}_{i, 1} (z), α_{i, 1} ({\hat{s}}_{i, 1})) d z) = \int_{t}^{\infty} K_{i, 1} ({\hat{s}}_{i, 1} (z), α_{i, 1}^{*} ({\hat{s}}_{i, 1})) d z, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} J^{*}_{i,1}(\hat{s}_{i,1})=\min _{\alpha _{i,1}\in \mathrm \Psi (\mathrm \Omega )} \Big (\int ^{\infty }_{t}K_{i,1}\big (\hat{s}_{i,1}(z),\alpha _{i,1}(\hat{s}_{i,1})\big )dz \Big )=\int ^{\infty }_{t}K_{i,1}\big (\hat{s}_{i,1}(z),\alpha ^{*}_{i,1}(\hat{s}_{i,1})\big )dz, \end{aligned} $$$

where $K_{i, 1} ({\hat{s}}_{i, 1}, α_{i, 1}) = {\hat{s}}_{i, 1}^{2} (t) + α_{i, 1}^{2} ({\hat{s}}_{i, 1})$ $Mathematical equation: $ K_{i,1}(\hat{s}_{i,1},\alpha_{i,1}) = \hat{s}^{2}_{i,1}(t)+\alpha^{2}_{i,1}(\hat{s}_{i,1}) $$ is the cost function. Under the optimal performance $J_{i, 1}^{*} ({\hat{s}}_{i, 1})$ $Mathematical equation: $ J^{*}_{i,1}(\hat{s}_{i,1}) $$ , treat the virtual tracking error as ${\hat{s}}_{i, 2} = 0$ $Mathematical equation: $ \hat{s}_{i,2}=0 $$ , i.e., view ${\hat{ζ}}_{i, 2} (t)$ $Mathematical equation: $ \hat{\zeta}_{i,2}(t) $$ as $α_{i, 1}^{*} ({\hat{s}}_{i, 1})$ $Mathematical equation: $ \alpha^{*}_{i,1}(\hat{s}_{i,1}) $$ . Then, the HJB equation is decomposed as

$\begin{matrix} H_{i, 1} ({\hat{s}}_{i, 1}, α_{i, 1}^{*}, \frac{\partial J_{i, 1}^{*}}{\partial {\hat{s}}_{i, 1}}) & = K_{i, 1} ({\hat{s}}_{i, 1}, α_{i, 1}^{*}) + \frac{\partial J_{i, 1}^{*} ({\hat{s}}_{i, 1})}{\partial {\hat{s}}_{i, 1}} {\dot{\hat{s}}}_{i, 1} \\ = {\hat{s}}_{i, 1}^{2} (t) + α_{i, 1}^{* 2} ({\hat{s}}_{i, 1}) + \frac{\partial J_{i, 1}^{*} ({\hat{s}}_{i, 1})}{\partial {\hat{s}}_{i, 1}} (α_{i, 1}^{*} ({\hat{s}}_{i, 1}) + \frac{g_{i, 1}}{ϖ_{i, 1}} - \frac{{\dot{ϖ}}_{i, 1} {\hat{η}}_{i, 1}}{ϖ_{i, 1}} \\ + \sum_{j = 1}^{N} a_{i, j} ({\hat{η}}_{i, 1} - {\hat{η}}_{j, 1})) \\ = 0 . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} H_{i,1}\left(\hat{s}_{i,1},\alpha ^{*}_{i,1},\frac{\partial J^{*}_{i,1}}{\partial \hat{s}_{i,1}}\right)&=K_{i,1}(\hat{s}_{i,1},\alpha ^{*}_{i,1}) +\frac{\partial J^{*}_{i,1}(\hat{s}_{i,1})}{\partial \hat{s}_{i,1}} \dot{\hat{s}}_{i,1}\nonumber \\&=\hat{s}^{2}_{i,1}(t)+\alpha ^{*2}_{i,1}(\hat{s}_{i,1}) +\frac{\partial J^{*}_{i,1}(\hat{s}_{i,1})}{\partial \hat{s}_{i,1}}\Big (\alpha ^{*}_{i,1}(\hat{s}_{i,1}) +\frac{g_{i,1}}{\varpi _{i,1}}-\frac{\dot{\varpi }_{i,1} \hat{\eta }_{i,1}}{\varpi _{i,1}}\nonumber \\&\qquad +\sum ^{N}_{j=1}a_{i,j}\big (\hat{\eta }_{i,1} -\hat{\eta }_{j,1}\big )\Big )\nonumber \\&=0. \end{aligned} $$$ (7)

By calculating $\frac{\partial H_{i, 1}}{\partial α_{i, 1}^{*}} = 0$ $Mathematical equation: $ \frac{\partial H_{i,1}}{\partial \alpha^{*}_{i,1}}=0 $$ , it established that $α_{i, 1}^{*} = - \frac{\partial J_{i, 1}^{*} ({\hat{s}}_{i, 1})}{2 \partial {\hat{s}}_{i, 1}}$ $Mathematical equation: $ \alpha^{*}_{i,1}=-\frac{{\partial J_{i,1}^*({{\hat s}_{i,1}})}}{{2\partial {{\hat s}_{i,1}}}} $$ . Then, the term $\frac{\partial J_{i, 1}^{*} ({\hat{s}}_{i, 1})}{\partial {\hat{s}}_{i, 1}}$ $Mathematical equation: $ \frac{\partial J^{*}_{i,1}(\hat{s}_{i,1})}{\partial \hat{s}_{i,1}} $$ can be decomposed as

$\begin{matrix} \frac{\partial J_{i, 1}^{*} ({\hat{s}}_{i, 1})}{\partial {\hat{s}}_{i, 1}} = (2 a_{i, 1} {\hat{s}}_{i, 1} - \frac{2 N (τ_{i, 1}) {\hat{ϑ}}_{i, 1} {\hat{s}}_{i, 1} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1} + ν_{i, 1}^{2}}} - \frac{2 N ({\overset{˘}{τ}}_{i, 1}) {\hat{ρ}}_{i, 1} {\hat{s}}_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} + ν_{i, 1}^{2}}} + J_{i, 1}^{0} ({\bar{\hat{η}}}_{i, 1}, {\hat{s}}_{i, 1})), \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \frac{\partial J^{*}_{i,1}(\hat{s}_{i,1})}{\partial \hat{s}_{i,1}}=\Big (2a_{i,1}\hat{s}_{i,1} -\frac{2N(\tau _{i,1})\hat{\vartheta }_{i,1} \hat{s}_{i,1}\bar{R}^{T}_{i,1}\bar{R}_{i,1}}{\sqrt{\hat{s}^{2}_{i,1}\bar{R}^{T}_{i,1}\bar{R}_{i,1} +\nu ^{2}_{i,1}}}-\frac{2N(\breve{\tau }_{i,1}) \hat{\rho }_{i,1}\hat{s}_{i,1}}{\sqrt{\hat{s}^{2}_{i,1} +\nu ^{2}_{i,1}}}+J^{0}_{i,1} (\bar{\hat{\eta }}_{i,1},\hat{s}_{i,1})\Big ), \end{aligned} $$$ (8)

where a_i, 1 is positive parameter, $N (τ_{i, 1}) = exp (τ_{i, 1}^{2}) sin (π τ_{i, 1})$ $Mathematical equation: $ N(\tau_{i,1})= \exp(\tau_{i,1}^{2})\sin(\pi\tau_{i,1}) $$ , $N ({\overset{˘}{τ}}_{i, 1}) = exp ({\overset{˘}{τ}}_{i, 1}^{2}) sin (π {\overset{˘}{τ}}_{i, 1})$ $Mathematical equation: $ N(\breve{\tau}_{i,1})= \exp(\breve{\tau}^{2}_{i,1})\sin(\pi\breve{\tau}_{i,1}) $$ , $ν_{i, 1} = δ_{i, 1} exp (- {\overset{˘}{δ}}_{i, 1} t)$ $Mathematical equation: $ \nu_{i,1}=\delta_{i,1}\exp(-\breve{\delta}_{i,1}t) $$ , δ_i, 1 and ${\overset{˘}{δ}}_{i, 1}$ $Mathematical equation: $ \breve{\delta}_{i,1} $$ are positive constants, ${\bar{R}}_{i, 1} \in R^{q_{1}}$ $Mathematical equation: $ \bar{R}_{i,1}\in R^{q_{1}} $$ represents the basis function vector, q₁ is the dimension of the input vector, $J_{i, 1}^{0} ({\bar{\hat{η}}}_{i, 1}, {\hat{s}}_{i, 1}) = \frac{\partial J_{i, 1}^{*} ({\hat{s}}_{i, 1})}{\partial {\hat{s}}_{i, 1}} - 2 a_{i, 1} {\hat{s}}_{i, 1} + \frac{2 N (τ_{i, 1}) {\hat{ϑ}}_{i, 1} {\hat{s}}_{i, 1} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1} + ν_{i, 1}^{2}}} + \frac{2 N ({\overset{˘}{τ}}_{i, 1}) {\hat{ρ}}_{i, 1} {\hat{s}}_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} + ν_{i, 1}^{2}}}$ $Mathematical equation: $ J^{0}_{i,1}(\bar{\hat{\eta}}_{i,1},\hat{s}_{i,1}) = \frac{\partial J^{*}_{i,1}(\hat{s}_{i,1})}{\partial \hat{s}_{i,1}} - 2a_{i,1}\hat{s}_{i,1} + \frac{2N(\tau_{i,1})\hat{\vartheta}_{i,1}\hat{s}_{i,1}\bar{R}^{T}_{i,1}\bar{R}_{i,1}} {\sqrt{\hat{s}^{2}_{i,1}\bar{R}^{T}_{i,1}\bar{R}_{i,1} + \nu^{2}_{i,1}}}+\frac{2N(\breve{\tau}_{i,1})\hat{\rho}_{i,1} \hat{s}_{i,1}}{\sqrt{\hat{s}^{2}_{i,1}+\nu^{2}_{i,1}}} $$ . Then, we have

$\begin{matrix} α_{i, 1}^{*} = - a_{i, 1} {\hat{s}}_{i, 1} + \frac{N (τ_{i, 1}) {\hat{ϑ}}_{i, 1} {\hat{s}}_{i, 1} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1} + ν_{i, 1}^{2}}} + \frac{N ({\overset{˘}{τ}}_{i, 1}) {\hat{ρ}}_{i, 1} {\hat{s}}_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} + ν_{i, 1}^{2}}} - \frac{1}{2} J_{i, 1}^{0} ({\bar{\hat{ζ}}}_{i, 1}, {\hat{s}}_{i, 1}) . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \alpha ^{*}_{i,1}=-a_{i,1}\hat{s}_{i,1} +\frac{N(\tau _{i,1})\hat{\vartheta }_{i,1}\hat{s}_{i,1} \bar{R}^{T}_{i,1}\bar{R}_{i,1}}{\sqrt{\hat{s}^{2}_{i,1} \bar{R}^{T}_{i,1}\bar{R}_{i,1}+\nu ^{2}_{i,1}}} +\frac{N(\breve{\tau }_{i,1})\hat{\rho }_{i,1}\hat{s}_{i,1}}{\sqrt{\hat{s}^{2}_{i,1}+\nu ^{2}_{i,1}}}-\frac{1}{2}J^{0}_{i,1} (\bar{\hat{\zeta }}_{i,1},\hat{s}_{i,1}). \end{aligned} $$$ (9)

Note that $J_{i, 1}^{0} ({\bar{\hat{ζ}}}_{i, 1}, {\hat{s}}_{i, 1})$ $Mathematical equation: $ J^{0}_{i,1}(\bar{\hat{\zeta}}_{i,1},\hat{s}_{i,1}) $$ is unknown. Thus, the RBF NNs are employed for approximate $J_{i, 1}^{0} ({\bar{\hat{ζ}}}_{i, 1}, {\hat{s}}_{i, 1})$ $Mathematical equation: $ J^{0}_{i,1}(\bar{\hat{\zeta}}_{i,1},\hat{s}_{i,1}) $$ . According to the introduction of RBF NNs in Section II, We have

$\begin{matrix} J_{i, 1}^{0} = W_{J i, 1}^{* T} {\bar{R}}_{J i, 1} ({\bar{\hat{ζ}}}_{i, 1}, {\hat{s}}_{i, 1}) + ϵ_{J i, 1} ({\bar{\hat{ζ}}}_{i, 1}, {\hat{s}}_{i, 1}), \end{matrix}$ $Mathematical equation: $$ \begin{aligned} J^{0}_{i,1}=W^{*T}_{Ji,1}\bar{R}_{Ji,1}(\bar{\hat{\zeta }}_{i,1}, \hat{s}_{i,1})+\epsilon _{Ji,1}(\bar{\hat{\zeta }}_{i,1}, \hat{s}_{i,1}), \end{aligned} $$$ (10)

where ϵ_J
i, 1 represents the approximation error.

Substituting (10) into (9) and (8), it follows that

$\begin{matrix} \frac{\partial J_{i, 1}^{*} ({\hat{s}}_{i, 1})}{\partial {\hat{s}}_{i, 1}} & = 2 a_{i, 1} {\hat{s}}_{i, 1} - \frac{2 N (τ_{i, 1}) {\hat{ϑ}}_{i, 1} {\hat{s}}_{i, 1} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1} + ν_{i, 1}^{2}}} - \frac{2 N ({\overset{˘}{τ}}_{i, 1}) {\hat{ρ}}_{i, 1} {\hat{s}}_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} + ν_{i, 1}^{2}}} + W_{J i, 1}^{* T} {\bar{R}}_{J i, 1} + ϵ_{J i, 1}, \\ α_{i, 1}^{*} & = - a_{i, 1} {\hat{s}}_{i, 1} + \frac{N (τ_{i, 1}) {\hat{ϑ}}_{i, 1} {\hat{s}}_{i, 1} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1} + ν_{i, 1}^{2}}} + \frac{N ({\frac{}{τ}}_{i, 1}) {\hat{ρ}}_{i, 1} {\hat{s}}_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} + ν_{i, 1}^{2}}} - \frac{1}{2} (W_{J i, 1}^{* T} {\bar{R}}_{J i, 1} + ϵ_{J i, 1}) . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \frac{\partial J^{*}_{i,1}(\hat{s}_{i,1})}{\partial \hat{s}_{i,1}}&= 2a_{i,1}\hat{s}_{i,1}-\frac{2N(\tau _{i,1})\hat{\vartheta }_{i,1} \hat{s}_{i,1}\bar{R}^{T}_{i,1}\bar{R}_{i,1}}{\sqrt{\hat{s}^{2}_{i,1}\bar{R}^{T}_{i,1}\bar{R}_{i,1} +\nu ^{2}_{i,1}}}-\frac{2N(\breve{\tau }_{i,1})\hat{\rho }_{i,1} \hat{s}_{i,1}}{\sqrt{\hat{s}^{2}_{i,1}+\nu ^{2}_{i,1}}} +W^{*T}_{Ji,1}\bar{R}_{Ji,1} + \epsilon _{Ji,1}, \\ \alpha _{i,1}^*&= -{a_{i,1}}{\hat{s}_{i,1}} + \frac{{N({\tau _{i,1}}){{\hat{\vartheta }}_{i,1}}{{\hat{s}}_{i,1}}\bar{R}_{i,1}^{T}{{\bar{R}}_{i,1}}}}{{\sqrt{\hat{s}_{i,1}^2\bar{R}_{i,1}^{T}{{\bar{R}}_{i,1}} + \nu _{i,1}^2}}} + \frac{{N({{{{} \over \tau }}_{i,1}}){{\hat{\rho }}_{i,1}}{{\hat{s}}_{i,1}}}}{{\sqrt{\hat{s}_{i,1}^2 + \nu _{i,1}^2}}} - \frac{1}{2}(W_{Ji,1}^{*T}{\bar{R}_{Ji,1}} + \epsilon _{Ji,1}). \end{aligned} $$$

Note that $\frac{\partial J_{i, 1}^{*} ({\hat{s}}_{i, 1})}{\partial {\hat{s}}_{i, 1}}$ $Mathematical equation: $ \frac{\partial J^{*}_{i,1}(\hat{s}_{i,1})}{\partial\hat{s}_{i,1}} $$ and $α_{i, 1}^{*}$ $Mathematical equation: $ \alpha^{*}_{i,1} $$ are unavailable due to the presence of the unknown ideal weight W_J
i, 1. To this end, the NN-based RL pave the way to tackle the difficulty.

The performance function under the critic NN and the optimal virtual control under the actor NN are designed as follows:

$\begin{matrix} \frac{\partial {\hat{J}}_{i, 1}^{*} ({\hat{s}}_{i, 1})}{\partial {\hat{s}}_{i, 1}} & = 2 a_{i, 1} {\hat{s}}_{i, 1} - \frac{2 N (τ_{i, 1}) {\hat{ϑ}}_{i, 1} {\hat{s}}_{i, 1} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1} + ν_{i, 1}^{2}}} - \frac{2 N ({\overset{˘}{τ}}_{i, 1}) {\hat{ρ}}_{i, 1} {\hat{s}}_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} + ν_{i, 1}^{2}}} + {\hat{W}}_{c i, 1}^{T} (t) {\bar{R}}_{J i, 1}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \frac{\partial \hat{J}^{*}_{i,1}(\hat{s}_{i,1})}{\partial \hat{s}_{i,1}}&=2a_{i,1}\hat{s}_{i,1}-\frac{2N(\tau _{i,1}) \hat{\vartheta }_{i,1}\hat{s}_{i,1}\bar{R}^{T}_{i,1} \bar{R}_{i,1}}{\sqrt{\hat{s}^{2}_{i,1}\bar{R}^{T}_{i,1}\bar{R}_{i,1} +\nu ^{2}_{i,1}}}-\frac{2N(\breve{\tau }_{i,1}) \hat{\rho }_{i,1}\hat{s}_{i,1}}{\sqrt{\hat{s}^{2}_{i,1} +\nu ^{2}_{i,1}}}+\hat{W}^{T}_{ci,1}(t) \bar{R}_{Ji,1},\end{aligned} $$$ (11)

$\begin{matrix} {\hat{α}}_{i, 1}^{*} & = - a_{i, 1} {\hat{s}}_{i, 1} + \frac{N (τ_{i, 1}) {\hat{ϑ}}_{i, 1} {\hat{s}}_{i, 1} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1} + ν_{i, 1}^{2}}} + \frac{N ({\overset{˘}{τ}}_{i, 1}) {\hat{ρ}}_{i, 1} {\hat{s}}_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} + ν_{i, 1}^{2}}} - \frac{1}{2} {\hat{W}}_{a i, 1}^{T} (t) {\bar{R}}_{J i, 1}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \hat{\alpha }^{*}_{i,1}&=-a_{i,1}\hat{s}_{i,1} +\frac{N(\tau _{i,1})\hat{\vartheta }_{i,1}\hat{s}_{i,1}\bar{R}^{T}_{i,1} \bar{R}_{i,1}}{\sqrt{\hat{s}^{2}_{i,1}\bar{R}^{T}_{i,1} \bar{R}_{i,1}+\nu ^{2}_{i,1}}}+\frac{N(\breve{\tau }_{i,1}) \hat{\rho }_{i,1}\hat{s}_{i,1}}{\sqrt{\hat{s}^{2}_{i,1} +\nu ^{2}_{i,1}}}-\frac{1}{2} \hat{W}^{T}_{ai,1}(t)\bar{R}_{Ji,1}, \end{aligned} $$$ (12)

where $\frac{\partial {\hat{J}}_{i, 1}^{*} ({\hat{s}}_{i, 1})}{\partial {\hat{s}}_{i, 1}}$ $Mathematical equation: $ \frac{\partial\hat{J}^{*}_{i,1}(\hat{s}_{i,1})}{\partial\hat{s}_{i,1}} $$ is the estimation of $\frac{\partial J_{i, 1}^{*} ({\hat{s}}_{i, 1})}{\partial {\hat{s}}_{i, 1}}$ $Mathematical equation: $ \frac{\partial J^{*}_{i,1}(\hat{s}_{i,1})}{\partial\hat{s}_{i,1}} $$ , ${\hat{α}}_{i, 1}^{*}$ $Mathematical equation: $ \hat{\alpha}^{*}_{i,1} $$ is identified as the estimate of $α_{i, 1}^{*}$ $Mathematical equation: $ \alpha^{*}_{i,1} $$ , the weights are symbolized as ${\hat{W}}_{c i, 1} (t) \in R^{q_{1}}$ $Mathematical equation: $ \hat{W}_{ci,1}(t)\in R^{q_{1}} $$ and ${\hat{W}}_{a i, 1} (t) \in R^{q_{1}}$ $Mathematical equation: $ \hat{W}_{ai,1}(t)\in R^{q_{1}} $$ .

The update laws ${\dot{\hat{W}}}_{c i, 1} (t)$ $Mathematical equation: $ \dot{\hat{W}}_{ci,1}(t) $$ and ${\dot{\hat{W}}}_{a i, 1} (t)$ $Mathematical equation: $ \dot{\hat{W}}_{ai,1}(t) $$ are set as

$\begin{matrix} \begin{matrix} {\begin{matrix} {\dot{\hat{W}}}_{c i, 1} (t) = & - ν_{i, 1} β_{c i, 1} {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} {\hat{W}}_{c i, 1} (t), \\ {\dot{\hat{W}}}_{a i, 1} (t) = & - {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} (ν_{i, 1} β_{a i, 1} ({\hat{W}}_{a i, 1} (t) - {\hat{W}}_{c i, 1} (t)) + ν_{i, 1} β_{c i, 1} {\hat{W}}_{c i, 1} (t)), \end{matrix} \end{matrix} \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \begin{split} \left\{ \begin{aligned} \dot{\hat{W}}_{ci,1}(t)=&-\nu _{i,1}\beta _{ci,1} \bar{R}^{T}_{Ji,1}\bar{R}_{Ji,1}\hat{W}_{ci,1}(t), \\ \dot{\hat{W}}_{ai,1}(t)=&-\bar{R}^{T}_{Ji,1} \bar{R}_{Ji,1}\Big (\nu _{i,1}\beta _{ai,1} \big (\hat{W}_{ai,1}(t)-\hat{W}_{ci,1}(t)\big ) +\nu _{i,1}\beta _{ci,1}\hat{W}_{ci,1}(t)\Big ), \end{aligned} \right. \end{split} \end{aligned} $$$ (13)

where β_c
i, 1 > 0 and β_a
i, 1 > 0 are the parameters, $β_{c i, 1} > \frac{1}{2}$ $Mathematical equation: $ \beta_{ci,1} > \frac{1}{2} $$ and $β_{a i, 1} > β_{c i, 1} > \frac{β_{a i, 1}}{2}$ $Mathematical equation: $ \beta_{ai,1} > \beta_{ci,1} > \frac{\beta_{ai,1}}{2} $$ .

By combining (7), (11) and (12), it yields that

$\begin{matrix} H_{i, 1} ({\hat{s}}_{i, 1}, {\hat{α}}_{i, 1}^{*}, \frac{\partial {\hat{J}}_{i, 1}^{*}}{\partial {\hat{s}}_{i, 1}}) = & {\hat{s}}_{i, 1}^{2} (t) + {\hat{α}}_{i, 1}^{* 2} ({\hat{s}}_{i, 1}) + \frac{\partial {\hat{J}}_{i, 1}^{*} ({\hat{s}}_{i, 1})}{\partial {\hat{s}}_{i, 1}} ({\hat{α}}_{i, 1}^{*} ({\hat{s}}_{i, 1}) + \frac{g_{i, 1}}{ϖ_{i, 1}} - \frac{{\dot{ϖ}}_{i, 1} {\hat{ζ}}_{i, 1}}{ϖ_{i, 1}} \\ + \sum_{j = 1}^{N} a_{i, j} ({\hat{ζ}}_{i, 1} - {\hat{ζ}}_{j, 1})) = 0 . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} H_{i,1}\Big (\hat{s}_{i,1},\hat{\alpha }^{*}_{i,1}, \frac{\partial \hat{J}^{*}_{i,1}}{\partial \hat{s}_{i,1}}\Big ) =&\hat{s}^{2}_{i,1}(t)+\hat{\alpha }^{*2}_{i,1}(\hat{s}_{i,1}) +\frac{\partial \hat{J}^{*}_{i,1}(\hat{s}_{i,1})}{\partial \hat{s}_{i,1}}\Big (\hat{\alpha }^{*}_{i,1}(\hat{s}_{i,1}) +\frac{g_{i,1}}{\varpi _{i,1}}-\frac{\dot{\varpi }_{i,1} \hat{\zeta }_{i,1}}{\varpi _{i,1}} \\&+\sum ^{N}_{j=1}a_{i,j}\big (\hat{\zeta }_{i,1} -\hat{\zeta }_{j,1}\big )\Big )=0. \end{aligned} $$$

Along with the fact that $H_{i, 1} ({\hat{s}}_{i, 1}, α_{i, 1}^{*}, \frac{\partial J_{i, 1}^{*}}{\partial {\hat{s}}_{i, 1}}) = 0$ $Mathematical equation: $ {H_{i,1}}({{\hat s}_{i,1}},\alpha_{i,1}^*,\frac{{\partial J_{i,1}^*}}{{\partial {{\hat s}_{i,1}}}})=0 $$ , the following residual error is obtained

$\begin{matrix} e_{i, 1} (t) = H_{i, 1} ({\hat{s}}_{i, 1}, {\hat{α}}_{i, 1}^{*}, \frac{\partial {\hat{J}}_{i, 1}^{*}}{\partial {\hat{s}}_{i, 1}}) - H_{i, 1} ({\hat{s}}_{i, 1}, α_{i, 1}^{*}, \frac{\partial J_{i, 1}^{*}}{\partial {\hat{s}}_{i, 1}}) = H_{i, 1} ({\hat{s}}_{i, 1}, {\hat{α}}_{i, 1}^{*}, \frac{\partial {\hat{J}}_{i, 1}^{*}}{\partial {\hat{s}}_{i, 1}}) . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} e_{i,1}(t) = H_{i,1}\Big (\hat{s}_{i,1},\hat{\alpha }^{*}_{i,1}, \frac{\partial \hat{J}^{*}_{i,1}}{\partial \hat{s}_{i,1}}\Big ) -H_{i,1}\Big (\hat{s}_{i,1},\alpha ^{*}_{i,1},\frac{\partial J^{*}_{i,1}}{\partial \hat{s}_{i,1}}\Big ) = H_{i,1}\Big (\hat{s}_{i,1}, \hat{\alpha }^{*}_{i,1},\frac{\partial \hat{J}^{*}_{i,1}}{\partial \hat{s}_{i,1}}\Big ). \end{aligned} $$$ (14)

Under the optimal control strategy ${\hat{α}}_{i, 1}^{*}$ $Mathematical equation: $ \hat{\alpha}^{*}_{i,1} $$ , e_i, 1(t) is anticipated to be zero, and yielding the following relationship.

$\begin{matrix} \frac{\partial H_{i, 1} ({\hat{s}}_{i, 1}, {\hat{α}}_{i, 1}^{*}, \frac{\partial {\hat{J}}_{i, 1}^{*} ({\hat{s}}_{i, 1})}{\partial {\hat{s}}_{i, 1}})}{\partial {\hat{W}}_{a i, 1}} = \frac{1}{2} {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} ({\hat{W}}_{a i, 1} (t) - {\hat{W}}_{c i, 1} (t)) = 0 . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \frac{\partial H_{i,1}\left(\hat{s}_{i,1},\hat{\alpha }^{*}_{i,1}, \frac{\partial \hat{J}^{*}_{i,1}(\hat{s}_{i,1})}{\partial \hat{s}_{i,1}}\right)}{\partial \hat{W}_{ai,1}} = \frac{1}{2}\bar{R}^{T}_{Ji,1}\bar{R}_{Ji,1} \Big (\hat{W}_{ai,1}(t)-\hat{W}_{ci,1}(t)\Big )=0. \end{aligned} $$$ (15)

In order to satisfy (15), a positive function can be defined as

$\begin{matrix} P_{i, 1} (t) = ({\hat{W}}_{a i, 1} (t) - {\hat{W}}_{c i, 1} (t))^{T} ({\hat{W}}_{a i, 1} (t) - {\hat{W}}_{c i, 1} (t)) . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} P_{i,1}(t)=\Big (\hat{W}_{ai,1}(t)-\hat{W}_{ci,1}(t)\Big )^{T} \Big (\hat{W}_{ai,1}(t)-\hat{W}_{ci,1}(t)\Big ). \end{aligned} $$$

It is noticeable that P_i, 1(t) is equal to (15) when P_i, 1(t)=0. By the derivative calculation, $\frac{\partial P_{i, 1} (t)}{\partial {\hat{W}}_{a i, 1} (t)} = - \frac{\partial P_{i, 1} (t)}{\partial {\hat{W}}_{c i, 1} (t)} = 2 ({\hat{W}}_{a i, 1} (t) - {\hat{W}}_{c i, 1} (t))$ $Mathematical equation: $ \frac{\partial P_{i,1}(t)}{\partial \hat{W}_{ai,1}(t)}=-\frac{\partial P_{i,1}(t)}{\partial \hat{W}_{ci,1}(t)}=2\Big(\hat{W}_{ai,1}(t)-\hat{W}_{ci,1}(t)\Big) $$ . According to (13), one can deduce

$\begin{matrix} \frac{d P_{i, 1} (t)}{dt} & = \frac{\partial P_{i, 1} (t)}{\partial {\hat{W}}_{c i, 1}^{T} (t)} {\dot{\hat{W}}}_{c i, 1} (t) + \frac{\partial P_{i, 1} (t)}{\partial {\hat{W}}_{a i, 1}^{T} (t)} {\dot{\hat{W}}}_{a i, 1} (t) \\ = - ν_{i, 1} β_{c i, 1} \frac{\partial P_{i, 1} (t)}{\partial {\hat{W}}_{c i, 1}^{T} (t)} {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} {\hat{W}}_{c i, 1} (t) - \frac{\partial P_{i, 1} (t)}{\partial {\hat{W}}_{a i, 1}^{T} (t)} {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} (ν_{i, 1} β_{a i, 1} ({\hat{W}}_{a i, 1} (t) \\ - {\hat{W}}_{c i, 1} (t)) + ν_{i, 1} β_{c i, 1} {\hat{W}}_{c i, 1} (t)) \\ = - \frac{1}{2} ν_{i, 1} β_{a i, 1} \frac{\partial P_{i, 1} (t)}{\partial {\hat{W}}_{a i, 1}^{T} (t)} \frac{\partial P_{i, 1} (t)}{\partial {\hat{W}}_{a i, 1} (t)} {\bar{R}}_{J i, 1}^{T} R_{J i, 1} \leq 0 . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \frac{dP_{i,1}(t)}{dt}&=\frac{\partial P_{i,1}(t)}{\partial \hat{W}^{T}_{ci,1}(t)}\dot{\hat{W}}_{ci,1}(t) +\frac{\partial P_{i,1}(t)}{\partial \hat{W}^{T}_{ai,1}(t)} \dot{\hat{W}}_{ai,1}(t) \\&=-\nu _{i,1}\beta _{ci,1}\frac{\partial P_{i,1}(t)}{\partial \hat{W}^{T}_{ci,1}(t)}\bar{R}^{T}_{Ji,1}\bar{R}_{Ji,1} \hat{W}_{ci,1}(t) - \frac{\partial P_{i,1}(t)}{\partial \hat{W}^{T}_{ai,1}(t)}\bar{R}^{T}_{Ji,1}\bar{R}_{Ji,1} \Big (\nu _{i,1}\beta _{ai,1}\big (\hat{W}_{ai,1}(t) \\&\qquad -\hat{W}_{ci,1}(t)\big )+\nu _{i,1}\beta _{ci,1} \hat{W}_{ci,1}(t)\Big ) \\&=-\frac{1}{2}\nu _{i,1}\beta _{ai,1}\frac{\partial P_{i,1}(t)}{\partial \hat{W}^{T}_{ai,1}(t)}\frac{\partial P_{i,1}(t)}{\partial \hat{W}_{ai,1}(t)}\bar{R}^{T}_{Ji,1}R_{Ji,1}\le 0. \end{aligned} $$$

According to the above analysis, (13) guarantee that P_i, 1(t)=0 can be achieved eventually. Furthermore, the optimal signal $α_{i, 1}^{*}$ $Mathematical equation: $ \alpha^{*}_{i,1} $$ can be derived.

Remark 3 The design of ${\dot{\hat{W}}}_{c i, 1} (t)$ $Mathematical equation: $ \dot{\hat{W}}_{ci,1}(t) $$ and ${\dot{\hat{W}}}_{a i, 1} (t)$ $Mathematical equation: $ \dot{\hat{W}}_{ai,1}(t) $$ is achieved via employing the negative gradient of the quadratic function P_i, 1(t), alleviating the computational burden compared to employing the gradient descent algorithm.

According to (1) and (6), ${\dot{s}}_{i, 1}$ $Mathematical equation: $ \dot{s}_{i,1} $$ is obtained by

$\begin{matrix} {\dot{s}}_{i, 1} = (s_{i, 2} + ϖ_{i, 1} {\hat{α}}_{i, 1}^{*} + g_{i, 1}) + ϖ_{i, 1} \sum_{j = 1}^{N} a_{i, j} ({\hat{ζ}}_{i, 1} - {\hat{ζ}}_{j, 1}) - {\dot{ϖ}}_{i, 1} {\hat{z}}_{i, 1} . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{s}_{i,1}=\big (s_{i,2}+\varpi _{i,1}\hat{\alpha }^{*}_{i,1} +g_{i,1}\big )+\varpi _{i,1}\sum _{j=1}^{N}a_{i,j} (\hat{\zeta }_{i,1}-\hat{\zeta }_{j,1}) -\dot{\varpi }_{i,1}\hat{z}_{i,1}. \end{aligned} $$$ (16)

Selected the Lyapunov function candidate as

$\begin{matrix} V_{i, 1} (t) = \frac{1}{2} s_{i, 1}^{2} + \frac{1}{2} {\tilde{θ}}_{i, 1}^{2} + \frac{1}{2} {\tilde{W}}_{c i, 1}^{T} (t) {\tilde{W}}_{c i, 1} (t) + \frac{1}{2} {\tilde{W}}_{a i, 1}^{T} (t) {\tilde{W}}_{a i, 1} (t) + \frac{1}{2} {\tilde{ρ}}_{i, 1}^{2}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} V_{i,1}(t)=\frac{1}{2}s^{2}_{i,1} +\frac{1}{2}\tilde{\theta }^{2}_{i,1} +\frac{1}{2}\tilde{W}^{T}_{ci,1}(t) \tilde{W}_{ci,1}(t)+\frac{1}{2}\tilde{W}^{T}_{ai,1} (t)\tilde{W}_{ai,1}(t)+\frac{1}{2}\tilde{\rho }^{2}_{i,1}, \end{aligned} $$$

where ${\tilde{W}}_{c i, 1} (t) = {\hat{W}}_{c i, 1} - W_{J i, 1}^{*}$ $Mathematical equation: $ \tilde{W}_{ci,1}(t)=\hat{W}_{ci,1}-W^{*}_{Ji,1} $$ and ${\tilde{W}}_{a i, 1} (t) = {\hat{W}}_{a i, 1} - W_{J i, 1}^{*}$ $Mathematical equation: $ \tilde{W}_{ai,1}(t)=\hat{W}_{ai,1}-W^{*}_{Ji,1} $$ .

The functional derivatives of V_i, 1(t) is computed along (16), which yields

$\begin{matrix} {\dot{V}}_{i, 1} (t) = & s_{i, 1} (s_{i, 2} + ϖ_{i, 1} {\hat{α}}_{i, 1}^{*}) + s_{i, 1} G_{i, 1} ({\bar{π}}_{i, 1}) + {\tilde{ϑ}}_{i, 1} {\dot{\hat{ϑ}}}_{i, 1} + {\tilde{W}}_{c i, 1}^{T} (t) {\dot{\hat{W}}}_{c i, 1} (t) \\ + {\tilde{W}}_{a i, 1}^{T} (t) {\dot{\hat{W}}}_{a i, 1} (t) + {\tilde{ρ}}_{i, 1} {\dot{\hat{ρ}}}_{i, 1}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{V}_{i,1}(t) =&s_{i,1}(s_{i,2}+\varpi _{i,1}\hat{\alpha }^{*}_{i,1}) +s_{i,1}G_{i,1}(\bar{\pi }_{i,1})+\tilde{\vartheta }_{i,1} \dot{\hat{\vartheta }}_{i,1}+\tilde{W}^{T}_{ci,1}(t) \dot{\hat{W}}_{ci,1}(t)\nonumber \\&+\tilde{W}^{T}_{ai,1}(t)\dot{\hat{W}}_{ai,1}(t) +\tilde{\rho }_{i,1}\dot{\hat{\rho }}_{i,1}, \end{aligned} $$$ (17)

where $G_{i, 1} ({\bar{π}}_{i, 1}) = g_{i, 1} + ϖ_{i, 1} \sum_{j = 1}^{N} a_{i, j} ({\hat{ζ}}_{i, 1} - {\hat{ζ}}_{j, 1}) - {\dot{ϖ}}_{i, 1} {\hat{z}}_{i, 1} - \frac{1}{2} s_{i, 1}$ $Mathematical equation: $ G_{i,1}(\bar{\pi}_{i,1}) = g_{i,1}+\varpi_{i,1}\sum_{j=1}^{N}a_{i,j}(\hat{\zeta}_{i,1}-\hat{\zeta}_{j,1}) -\dot{\varpi}_{i,1}\hat{z}_{i,1}-\frac{1}{2}s_{i,1} $$ and ${\bar{π}}_{i, 1} = {[{\hat{ζ}}_{i, 1}, {\hat{ζ}}_{j, 1}]}^{T}$ $Mathematical equation: $ \bar{\pi}_{i,1}=[\hat{\zeta}_{i,1},\hat{\zeta}_{j,1}]^{T} $$ .

The RBF NNs $ξ_{i, 1}^{* T} {\bar{R}}_{i, 1} ({\bar{π}}_{i, 1})$ $Mathematical equation: $ \xi^{\ast T}_{i,1}\bar{R}_{i,1}(\bar{\pi}_{i,1}) $$ are utilized to approximate $G_{i, 1} ({\bar{π}}_{i, 1})$ $Mathematical equation: $ G_{i,1}(\bar{\pi}_{i,1}) $$ . Then, we have

$\begin{matrix} s_{i, 1} G_{i, 1} ({\bar{π}}_{i, 1}) = s_{i, 1} (ξ_{i, 1}^{* T} {\bar{R}}_{i, 1} ({\bar{π}}_{i, 1}) + ϵ_{i, 1} ({\bar{π}}_{i, 1})) \leq \frac{θ_{i, 1} {\hat{s}}_{i, 1}^{2} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1} + ν_{i, 1}^{2}}} + ϑ_{i, 1} ν_{i, 1} + s_{i, 1} ϵ_{i, 1}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} s_{i,1}G_{i,1}(\bar{\pi }_{i,1})=s_{i,1}\Big (\xi _{i,1}^{*T} \bar{R}_{i,1}(\bar{\pi }_{i,1})+\epsilon _{i,1}(\bar{\pi }_{i,1})\Big ) \le \frac{\theta _{i,1}\hat{s}^{2}_{i,1}\bar{R}^{T}_{i,1}\bar{R}_{i,1}}{\sqrt{\hat{s}^{2}_{i,1}\bar{R}^{T}_{i,1}\bar{R}_{i,1} +\nu ^{2}_{i,1}}}+\vartheta _{i,1}\nu _{i,1}+s_{i,1}\epsilon _{i,1}, \end{aligned} $$$ (18)

where |ϵ_i, 1|≤ε_i, 1. By applying Young’s inequality, the unknown parameter vector $ξ_{i, 1}^{*}$ $Mathematical equation: $ \xi^{\ast}_{i,1} $$ is lumped into a scalar such that $ξ_{i, 1}^{* T} ξ_{i, 1}^{*} = θ_{i, 1}$ $Mathematical equation: $ \xi^{ \ast T}_{i,1}\xi^{\ast}_{i,1} = \theta_{i,1} $$ . This significantly reduces the number of parameters to be estimated.

Considering the optimal virtual controller (12) with the (18), from (17), we derived

$\begin{matrix} {\dot{V}}_{i, 1} (t) \leq & s_{i, 1} ϖ_{i, 1} (- a_{i, 1} {\hat{s}}_{i, 1} + \frac{N (τ_{i, 1}) {\hat{ϑ}}_{i, 1} {\hat{s}}_{i, 1} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1} + ν_{i, 1}^{2}}} + \frac{N ({\overset{˘}{τ}}_{i, 1}) {\hat{ρ}}_{i, 1} {\hat{s}}_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} + ν_{i, 1}^{2}}} - \frac{1}{2} {\hat{W}}_{a i, 1}^{T} (t) {\bar{R}}_{J i, 1}) \\ + \frac{ϑ_{i, 1} {\hat{s}}_{i, 1}^{2} {\bar{R}}_{i, 1}^{T} R_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1} + ν_{i, 1}^{2}}} + s_{i, 1} ϵ_{i, 1} + ϑ_{i, 1} ν_{i, 1} + {\tilde{ϑ}}_{i, 1} {\dot{\hat{ϑ}}}_{i, 1} + {\tilde{ρ}}_{i, 1} {\dot{\hat{ρ}}}_{i, 1} + {\tilde{W}}_{c i, 1}^{T} (t) {\dot{\hat{W}}}_{c i, 1} (t) \\ + {\tilde{W}}_{a i, 1}^{T} (t) {\dot{\hat{W}}}_{a i, 1} (t) + s_{i, 1} s_{i, 2} + \frac{1}{2} s_{i, 1}^{2} . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{V}_{i,1}(t)\le &s_{i,1}\varpi _{i,1}\Big (-a_{i,1} \hat{s}_{i,1}+\frac{N(\tau _{i,1})\hat{\vartheta }_{i,1}\hat{s}_{i,1} \bar{R}^{T}_{i,1}\bar{R}_{i,1}}{\sqrt{\hat{s}^{2}_{i,1} \bar{R}^{T}_{i,1}\bar{R}_{i,1}+\nu ^{2}_{i,1}}}+\frac{N(\breve{\tau }_{i,1}) \hat{\rho }_{i,1}\hat{s}_{i,1}}{\sqrt{\hat{s}^{2}_{i,1}+\nu ^{2}_{i,1}}} -\frac{1}{2}\hat{W}^{T}_{ai,1}(t) \bar{R}_{Ji,1}\Big ) \nonumber \\&+\frac{\vartheta _{i,1}\hat{s}^{2}_{i,1}\bar{R}^{T}_{i,1}R_{i,1}}{\sqrt{\hat{s}^{2}_{i,1}\bar{R}^{T}_{i,1}\bar{R}_{i,1} +\nu ^{2}_{i,1}}}+s_{i,1}\epsilon _{i,1}+\vartheta _{i,1}\nu _{i,1} +\tilde{\vartheta }_{i,1}\dot{\hat{\vartheta }}_{i,1}+\tilde{\rho }_{i,1} \dot{\hat{\rho }}_{i,1}+\tilde{W}^{T}_{ci,1}(t) \dot{\hat{W}}_{ci,1}(t)\nonumber \\&+\tilde{W}^{T}_{ai,1}(t)\dot{\hat{W}}_{ai,1}(t) +s_{i,1}s_{i,2}+\frac{1}{2}s^{2}_{i,1}. \end{aligned} $$$ (19)

By employing Young’s inequality, it is deduced that

$\begin{matrix} \begin{matrix} {\begin{matrix} s_{i, 1} s_{i, 2} & \leq \frac{1}{2} (s_{i, 2}^{2} + s_{i, 1}^{2}), \\ | s_{i, 1} | (ε_{i, 1} & + | \frac{1}{2} ϖ_{i, 1} {\hat{W}}_{a i, 1}^{T} (t) {\bar{R}}_{J i, 1} |) \leq \frac{ρ_{i, 1} {\hat{s}}_{i, 1}^{2}}{\sqrt{{\hat{s}}_{i, 1}^{2} + ν_{i, 1}^{2}}} + ρ_{i, 1} ν_{i, 1} . \end{matrix} \end{matrix} \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \begin{split} \left\{ \begin{aligned} s_{i,1}s_{i,2}&\le \frac{1}{2}\Big (s^{2}_{i,2}+s^{2}_{i,1}\Big ),\\ \vert s_{i,1}\vert \Big (\varepsilon _{i,1}&+\vert \frac{1}{2}\varpi _{i,1}\hat{W}^{T}_{ai,1}(t) \bar{R}_{Ji,1}\vert \Big )\le \frac{\rho _{i,1}\hat{s}^{2}_{i,1}}{\sqrt{\hat{s}^{2}_{i,1}+\nu ^{2}_{i,1}}}+\rho _{i,1}\nu _{i,1}. \end{aligned} \right. \end{split} \end{aligned} $$$ (20)

The adaptive laws are formulated as follows to estimate the sensor bias fault

$\begin{matrix} {\dot{\hat{ϑ}}}_{i, 1} = \frac{{\hat{s}}_{i, 1}^{2} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1} + ν_{i, 1}^{2}}} - ν_{i, 1} {\hat{ϑ}}_{i, 1}, {\dot{\hat{ρ}}}_{i, 1} = \frac{{\hat{s}}_{i, 1}^{2}}{\sqrt{{\hat{s}}_{i, 1}^{2} + ν_{i, 1}^{2}}} - ν_{i, 1} {\hat{ρ}}_{i, 1} . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{\hat{\vartheta }}_{i,1}=\frac{\hat{s}^{2}_{i,1} \bar{R}^{T}_{i,1}\bar{R}_{i,1}}{\sqrt{\hat{s}^{2}_{i,1}\bar{R}^{T}_{i,1}\bar{R}_{i,1} +\nu ^{2}_{i,1}}}-\nu _{i,1}\hat{\vartheta }_{i,1}, \qquad \dot{\hat{\rho }}_{i,1}=\frac{\hat{s}^{2}_{i,1}}{\sqrt{\hat{s}^{2}_{i,1} +\nu ^{2}_{i,1}}}-\nu _{i,1}\hat{\rho }_{i,1}. \end{aligned} $$$ (21)

According to (13) and (19)–(21), it yields

$\begin{matrix} {\dot{V}}_{i, 1} (t) \leq & s_{i, 1}^{2} + \frac{1}{2} s_{i, 2}^{2} - a_{i, 1} s_{i, 1}^{2} + (N (τ_{i, 1}) ϖ_{i, 1}^{2} + 1) {\dot{τ}}_{i, 1} + (N ({\overset{˘}{τ}}_{i, 1}) ϖ_{i, 1}^{2} + 1) {\dot{\overset{˘}{τ}}}_{i, 1} - ν_{i, 1} {\tilde{ϑ}}_{i, 1} {\hat{ϑ}}_{i, 1} \\ - ν_{i, 1} {\tilde{ρ}}_{i, 1} {\hat{ρ}}_{i, 1} + θ_{i, 1} ν_{i, 1} + ρ_{i, 1} ν_{i, 1} - {\tilde{W}}_{c i, 1}^{T} (t) ν_{i, 1} β_{c i, 1} {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} {\hat{W}}_{c i, 1} (t) \\ - {\tilde{W}}_{a i, 1}^{T} (t) {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} (ν_{i, 1} β_{a i, 1} ({\hat{W}}_{a i, 1} (t) - {\hat{W}}_{c i, 1} (t)) + ν_{i, 1} β_{c i, 1} {\hat{W}}_{c i, 1} (t)), \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{V}_{i,1}(t)\le &s^{2}_{i,1}+\frac{1}{2}s^{2}_{i,2} -a_{i,1}s^{2}_{i,1}+\big (N(\tau _{i,1})\varpi ^{2}_{i,1} +1\big )\dot{\tau }_{i,1}+\big (N(\breve{\tau }_{i,1}) \varpi _{i,1}^{2}+1\big )\dot{\breve{\tau }}_{i,1} -\nu _{i,1}\tilde{\vartheta }_{i,1} \hat{\vartheta }_{i,1}\nonumber \\&-\nu _{i,1}\tilde{\rho }_{i,1}\hat{\rho }_{i,1} +\theta _{i,1}\nu _{i,1}+\rho _{i,1}\nu _{i,1} -\tilde{W}^{T}_{ci,1}(t)\nu _{i,1}\beta _{ci,1}\bar{R}^{T}_{Ji,1} \bar{R}_{Ji,1}\hat{W}_{ci,1}(t)\nonumber \\&-\tilde{W}^{T}_{ai,1}(t)\bar{R}^{T}_{Ji,1}\bar{R}_{Ji,1} \Big (\nu _{i,1}\beta _{ai,1}\big (\hat{W}_{ai,1}(t) -\hat{W}_{ci,1}(t)\big )+\nu _{i,1}\beta _{ci,1}\hat{W}_{ci,1}(t)\Big ), \end{aligned} $$$ (22)

where ${\dot{τ}}_{i, 1} = \frac{{\hat{ϑ}}_{i, 1} {\hat{s}}_{i, 1}^{2} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1}}{\sqrt{{\hat{s}}_{i, 1}^{2} {\bar{R}}_{i, 1}^{T} {\bar{R}}_{i, 1} + ν_{i, 1}^{2}}}$ $Mathematical equation: $ \dot{\tau}_{i,1} = \frac{\hat{\vartheta}_{i,1}\hat{s}^{2}_{i,1}\bar{R}^{T}_{i,1} \bar{R}_{i,1}}{\sqrt{\hat{s}^{2}_{i,1} \bar{R}^{T}_{i,1}\bar{R}_{i,1} + \nu^{2}_{i,1}}} $$ and ${\dot{\overset{˘}{τ}}}_{i, 1} = \frac{{\hat{ρ}}_{i, 1} {\hat{s}}_{i, 1}^{2}}{\sqrt{{\hat{s}}_{i, 1}^{2} + ν_{i, 1}^{2}}}$ $Mathematical equation: $ \dot{\breve{\tau}}_{i,1} = \frac{\hat{\rho}_{i,1}\hat{s}^{2}_{i,1}} {\sqrt{\hat{s}^{2}_{i,1} + \nu^{2}_{i,1}}} $$ .

By using ${\tilde{W}}_{c i, 1} (t) = {\hat{W}}_{c i, 1} - W_{J i, 1}^{*}$ $Mathematical equation: $ \tilde{W}_{ci,1}(t)=\hat{W}_{ci,1}-W^{*}_{Ji,1} $$ and ${\tilde{W}}_{a i, 1} (t) = {\hat{W}}_{a i, 1} - W_{J i, 1}^{*}$ $Mathematical equation: $ \tilde{W}_{ai,1}(t)=\hat{W}_{ai,1}-W^{*}_{Ji,1} $$ , it follows that

$\begin{matrix} \begin{matrix} {\begin{matrix} {\tilde{W}}_{c i, 1}^{T} (t) & {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} {\hat{W}}_{c i, 1} (t) \\ = \frac{1}{2} {\tilde{W}}_{c i, 1}^{T} (t) {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} {\tilde{W}}_{c i, 1} (t) + \frac{1}{2} {\hat{W}}_{c i, 1}^{T} (t) {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} {\hat{W}}_{c i, 1} (t) - \frac{1}{2} (W_{J i, 1}^{* T} {\bar{R}}_{J i, 1})^{2}, \\ {\tilde{W}}_{a i, 1}^{T} (t) & {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} {\hat{W}}_{a i, 1} (t) \\ = \frac{1}{2} {\tilde{W}}_{a i, 1}^{T} (t) {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} {\tilde{W}}_{a i, 1} (t) + \frac{1}{2} {\hat{W}}_{a i, 1}^{T} (t) {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} {\hat{W}}_{a i, 1} (t) - \frac{1}{2} (W_{J i, 1}^{* T} {\bar{R}}_{J i, 1})^{2} . \end{matrix} \end{matrix} \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \begin{split} \left\{ \begin{aligned} \tilde{W}^{T}_{ci,1}(t)&\bar{R}^{T}_{Ji,1}\bar{R}_{Ji,1}\hat{W}_{ci,1}(t) \\&=\frac{1}{2}\tilde{W}^{T}_{ci,1}(t)\bar{R}^{T}_{Ji,1}\bar{R}_{Ji,1} \tilde{W}_{ci,1}(t)+\frac{1}{2}\hat{W}^{T}_{ci,1}(t)\bar{R}^{T}_{Ji,1} \bar{R}_{Ji,1}\hat{W}_{ci,1}(t)-\frac{1}{2}\big (W^{*T}_{Ji,1}\bar{R}_{Ji,1}\big )^2, \\ \tilde{W}^{T}_{ai,1}(t)&\bar{R}^{T}_{Ji,1}\bar{R}_{Ji,1}\hat{W}_{ai,1}(t) \\&=\frac{1}{2}\tilde{W}^{T}_{ai,1}(t)\bar{R}^{T}_{Ji,1}\bar{R}_{Ji,1} \tilde{W}_{ai,1}(t)+\frac{1}{2}\hat{W}^{T}_{ai,1}(t)\bar{R}^{T}_{Ji,1} \bar{R}_{Ji,1}\hat{W}_{ai,1}(t)-\frac{1}{2}\big (W^{*T}_{Ji,1}\bar{R}_{Ji,1}\big )^{2}. \end{aligned} \right. \end{split} \end{aligned} $$$ (23)

Utilizing the Young’s inequality scaling, it decomposes as

$\begin{matrix} \begin{matrix} {\begin{matrix} - ν_{i, 1} {\tilde{ϑ}}_{i, 1} {\hat{ϑ}}_{i, 1} & \leq \frac{1}{2} ν_{i, 1} ϑ_{i, 1}^{2} - \frac{1}{2} ν_{i, 1} {\tilde{ϑ}}_{i, 1}^{2}, \\ - ν_{i, 1} {\tilde{ρ}}_{i, 1} {\hat{ρ}}_{i, 1} & \leq \frac{1}{2} ν_{i, 1} ρ_{i, 1}^{2} - \frac{1}{2} ν_{i, 1} {\tilde{ρ}}_{i, 1}^{2}, \end{matrix} \end{matrix} \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \begin{split} \left\{ \begin{aligned} -\nu _{i,1}\tilde{\vartheta }_{i,1}\hat{\vartheta }_{i,1}&\le \frac{1}{2}\nu _{i,1}\vartheta ^{2}_{i,1}-\frac{1}{2}\nu _{i,1} \tilde{\vartheta }^{2}_{i,1},\\ -\nu _{i,1}\tilde{\rho }_{i,1}\hat{\rho }_{i,1}&\le \frac{1}{2}\nu _{i,1}\rho ^{2}_{i,1}-\frac{1}{2}\nu _{i,1}\tilde{\rho }^{2}_{i,1}, \end{aligned} \right. \end{split} \end{aligned} $$$ (24)

and

$\begin{matrix} (β_{a i, 1} & - β_{c i, 1}) {\tilde{W}}_{a i, 1}^{T} (t) {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} {\hat{W}}_{c i, 1} (t) \\ \leq \frac{(β_{a i, 1} - β_{c i, 1})}{2} ({\tilde{W}}_{a i, 1}^{T} (t) {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} {\tilde{W}}_{a i, 1} (t) + {\hat{W}}_{c i, 1}^{T} (t) {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} {\hat{W}}_{c i, 1} (t)) . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} (\beta _{ai,1}&-\beta _{ci,1})\tilde{W}^{T}_{ai,1}(t)\bar{R}^{T}_{Ji,1} \bar{R}_{Ji,1}\hat{W}_{ci,1}(t) \nonumber \\&\le \frac{(\beta _{ai,1}-\beta _{ci,1})}{2}\Big (\tilde{W}^{T}_{ai,1}(t) \bar{R}^{T}_{Ji,1}\bar{R}_{Ji,1}\tilde{W}_{ai,1}(t)+\hat{W}^{T}_{ci,1}(t) \bar{R}^{T}_{Ji,1}\bar{R}_{Ji,1}\hat{W}_{ci,1}(t)\Big ). \end{aligned} $$$ (25)

Substituting (23)–(25) into (22), it yields

$\begin{matrix} {\dot{V}}_{i, 1} (t) \leq & \frac{1}{2} s_{i, 2}^{2} - (a_{i, 1} - 1) s_{i, 1}^{2} - \frac{1}{2} ν_{i, 1} {\tilde{ρ}}_{i, 1}^{2} - \frac{1}{2} ν_{i, 1} {\tilde{ϑ}}_{i, 1}^{2} - \frac{ν_{i, 1} β_{c i, 1}}{2} {\tilde{W}}_{c i, 1}^{T} {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} {\tilde{W}}_{c i, 1} \\ - \frac{ν_{i, 1} β_{c i, 1}}{2} {\tilde{W}}_{a i, 1}^{T} {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} {\tilde{W}}_{a i, 1} + \frac{1}{2} ν_{i, 1} ϑ_{i, 1}^{2} + (N (τ_{i, 1}) ϖ_{i, 1}^{2} + 1) {\dot{τ}}_{i, 1} + (N ({\overset{˘}{τ}}_{i, 1}) ϖ_{i, 1}^{2} + 1) {\dot{\overset{˘}{τ}}}_{i, 1} \\ - \frac{ν_{i, 1} β_{a i, 1}}{2} {\hat{W}}_{a i, 1}^{T} {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} {\hat{W}}_{a i, 1} + \frac{1}{2} ν_{i, 1} ρ_{i, 1}^{2} + (\frac{ν_{i, 1} β_{a i, 1}}{2} - ν_{i, 1} β_{c i, 1}) {\hat{W}}_{c i, 1}^{T} {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} {\hat{W}}_{c i, 1} \\ + (\frac{ν_{i, 1} β_{c i, 1}}{2} + \frac{ν_{i, 1} β_{a i, 1}}{2}) (W_{J i, 1}^{* T} {\bar{R}}_{J i, 1})^{2} + θ_{i, 1} ν_{i, 1} + ρ_{i, 1} ν_{i, 1} \\ \leq & \frac{1}{2} s_{i, 2}^{2} - (a_{i, 1} - 1) s_{i, 1}^{2} - \frac{ν_{i, 1} {\tilde{θ}}_{i, 1}^{2}}{2} - \frac{ν_{i, 1} {\tilde{ρ}}_{i, 1}^{2}}{2} + (N (τ_{i, 1}) ϖ_{i, 1}^{2} + 1) {\dot{τ}}_{i, 1} + (N ({\overset{˘}{τ}}_{i, 1}) ϖ_{i, 1}^{2} + 1) {\dot{\overset{˘}{τ}}}_{i, 1} \\ - \frac{ν_{i, 1} β_{c i, 1}}{2} {\tilde{W}}_{c i, 1}^{T} {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} {\tilde{W}}_{c i, 1} - \frac{ν_{i, 1} β_{c i, 1}}{2} {\tilde{W}}_{a i, 1}^{T} {\bar{R}}_{J i, 1}^{T} {\bar{R}}_{J i, 1} {\tilde{W}}_{a i, 1} + μ_{i, 1}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{V}_{i,1}(t)\le &\frac{1}{2}s^{2}_{i,2}-(a_{i,1} -1)s^{2}_{i,1}-\frac{1}{2}\nu _{i,1}\tilde{\rho }^{2}_{i,1} -\frac{1}{2}\nu _{i,1}\tilde{\vartheta }^{2}_{i,1}-\frac{\nu _{i,1} \beta _{ci,1}}{2}\tilde{W}^{T}_{ci,1}\bar{R}^{T}_{Ji,1} \bar{R}_{Ji,1}\tilde{W}_{ci,1} \nonumber \\&-\frac{\nu _{i,1}\beta _{ci,1}}{2}\tilde{W}^{T}_{ai,1} \bar{R}^{T}_{Ji,1}\bar{R}_{Ji,1}\tilde{W}_{ai,1}+\frac{1}{2}\nu _{i,1} \vartheta ^{2}_{i,1}+\big (N(\tau _{i,1})\varpi _{i,1}^{2}+1\big ) \dot{\tau }_{i,1}+\big (N(\breve{\tau }_{i,1})\varpi _{i,1}^{2}+1\big ) \dot{\breve{\tau }}_{i,1} \nonumber \\&-\frac{\nu _{i,1}\beta _{ai,1}}{2}\hat{W}^{T}_{ai,1}\bar{R}^{T}_{Ji,1} \bar{R}_{Ji,1}\hat{W}_{ai,1}+\frac{1}{2}\nu _{i,1} \rho ^{2}_{i,1}+\left(\frac{\nu _{i,1}\beta _{ai,1}}{2}-\nu _{i,1} \beta _{ci,1}\right)\hat{W}^{T}_{ci,1}\bar{R}^{T}_{Ji,1} \bar{R}_{Ji,1}\hat{W}_{ci,1} \nonumber \\&+\left(\frac{\nu _{i,1}\beta _{ci,1}}{2}+\frac{\nu _{i,1} \beta _{ai,1}}{2}\right)\big (W^{*T}_{Ji,1}\bar{R}_{Ji,1}\big )^{2} +\theta _{i,1}\nu _{i,1}+\rho _{i,1}\nu _{i,1} \nonumber \\ \le &\frac{1}{2}s^{2}_{i,2}-(a_{i,1}-1)s^{2}_{i,1} -\frac{\nu _{i,1}\tilde{\theta }^{2}_{i,1}}{2}-\frac{\nu _{i,1} \tilde{\rho }^{2}_{i,1}}{2}+\big (N(\tau _{i,1})\varpi _{i,1}^{2} +1\big )\dot{\tau }_{i,1}+\big (N(\breve{\tau }_{i,1})\varpi _{i,1}^{2} +1\big )\dot{\breve{\tau }}_{i,1} \nonumber \\&-\frac{\nu _{i,1}\beta _{ci,1}}{2}\tilde{W}^{T}_{ci,1} \bar{R}^{T}_{Ji,1}\bar{R}_{Ji,1}\tilde{W}_{ci,1}-\frac{\nu _{i,1} \beta _{ci,1}}{2}\tilde{W}^{T}_{ai,1}\bar{R}^{T}_{Ji,1} \bar{R}_{Ji,1}\tilde{W}_{ai,1}+\mu _{i,1}, \end{aligned} $$$ (26)

where $μ_{i, 1} = \frac{1}{2} ν_{i, 1} θ_{i, 1}^{2} + \frac{1}{2} ν_{i, 1} ρ_{i, 1}^{2} + (\frac{ν_{i, 1} β_{c i, 1}}{2} + \frac{ν_{i, 1} β_{a i, 1}}{2}) (W_{J i, 1}^{* T} {\bar{R}}_{J i, 1})^{2} + θ_{i, 1} ν_{i, 1} + ρ_{i, 1} ν_{i, 1}$ $Mathematical equation: $ \mu_{i,1} = \frac{1}{2}\nu_{i,1}\theta^{2}_{i,1} + \frac{1}{2}\nu_{i,1}\rho^{2}_{i,1} + \big(\frac{\nu_{i,1}\beta_{ci,1}}{2} + \frac{\nu_{i,1}\beta_{ai,1}}{2}\big)\big(W^{*T}_{Ji,1} \bar{R}_{Ji,1}\big)^{2}+\theta_{i,1}\nu_{i,1}+\rho_{i,1}\nu_{i,1} $$ .

Step l (l = 2, …, m − 1): According to (5), ${\dot{\hat{s}}}_{i, l}$ $Mathematical equation: $ \dot{\hat{s}}_{i,l} $$ is computed as

$\begin{matrix} {\dot{\hat{s}}}_{i, l} = {\hat{ζ}}_{i, l + 1} + \frac{g_{i, l}}{ϖ_{i, l}} - \frac{{\dot{ϖ}}_{i, l} {\hat{x}}_{i, l}}{ϖ_{i, l}} - {\dot{\hat{α}}}_{i, l - 1}^{*} . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{\hat{s}}_{i,l} = \hat{\zeta }_{i,l+1} + \frac{g_{i,l}}{\varpi _{i,l}}-\frac{\dot{\varpi }_{i,l} \hat{x}_{i,l}}{\varpi _{i,l}}-\dot{\hat{\alpha }}^{*}_{i,l-1}. \end{aligned} $$$

The optimal cost function corresponding to $α_{i, l}^{*} ({\hat{s}}_{i, l})$ $Mathematical equation: $ \alpha^{*}_{i,l}(\hat{s}_{i,l}) $$ is expressed as

$\begin{matrix} J_{i, l}^{*} ({\hat{s}}_{i, l}) = min_{α_{i, l} \in Ψ (Ω)} (\int_{t}^{\infty} K_{i, l} ({\hat{s}}_{i, l} (z), α_{i, 1} ({\hat{s}}_{i, l})) d z) = \int_{t}^{\infty} K_{i, l} ({\hat{s}}_{i, l} (z), α_{i, l}^{*} ({\hat{s}}_{i, l})) d z, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} J^{*}_{i,l}(\hat{s}_{i,l})=\min _{\alpha _{i,l}\in \mathrm \Psi (\mathrm \Omega )} \left(\int ^{\infty }_{t}K_{i,l}\big (\hat{s}_{i,l}(z), \alpha _{i,1}(\hat{s}_{i,l})\big )dz\right) = \int ^{\infty }_{t}K_{i,l}\big (\hat{s}_{i,l}(z), \alpha ^{*}_{i,l}(\hat{s}_{i,l})\big )dz,\nonumber \end{aligned} $$$

where $K_{i, l} ({\hat{s}}_{i, l}, α_{i, l}) = {\hat{s}}_{i, l}^{2} (t) + α_{i, l}^{2}$ $Mathematical equation: $ K_{i,l}(\hat{s}_{i,l},\alpha_{i,l}) = \hat{s}^{2}_{i,l}(t)+\alpha^{2}_{i,l} $$ .

The approach similar to the previous step can further derive the following HJB equation.

$\begin{matrix} H_{i, l} ({\hat{s}}_{i, l}, α_{i, l}^{*}, \frac{\partial J_{i, l}^{*}}{\partial {\hat{s}}_{i, l}}) & = {\hat{s}}_{i, l}^{2} (t) + α_{i, l}^{* 2} ({\hat{s}}_{i, l}) + \frac{\partial J_{i}^{*} ({\hat{s}}_{i, l})}{\partial {\hat{s}}_{i, l}} (α_{i, l}^{*} ({\hat{s}}_{i, l}) + \frac{g_{i, l}}{ϖ_{i, l}} - \frac{{\dot{ϖ}}_{i, l} {\hat{η}}_{i, l}}{ϖ_{i, l}} - {\dot{\hat{α}}}_{i, l - 1}^{*}) \\ = 0 . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} H_{i,l}\left(\hat{s}_{i,l},\alpha ^{*}_{i,l},\frac{\partial J^{*}_{i,l}}{\partial \hat{s}_{i,l}}\right)&=\hat{s}^{2}_{i,l}(t)+\alpha ^{*2}_{i,l} (\hat{s}_{i,l})+\frac{\partial J^{*}_{i}(\hat{s}_{i,l})}{\partial \hat{s}_{i,l}}\Big (\alpha ^{*}_{i,l}(\hat{s}_{i,l})+\frac{g_{i,l}}{\varpi _{i,l}}-\frac{\dot{\varpi }_{i,l}\hat{\eta }_{i,l}}{\varpi _{i,l}} -\dot{\hat{\alpha }}^{*}_{i,l-1}\Big ) \\&=0. \end{aligned} $$$

By solving $\frac{\partial H_{i, l}}{\partial α_{i, l}^{*}} = 0$ $Mathematical equation: $ \frac{\partial H_{i,l}}{\partial \alpha^{*}_{i,l}}=0 $$ , the relationship that relates $α_{i, l}^{*}$ $Mathematical equation: $ \alpha^{*}_{i,l} $$ to $\frac{\partial J_{i, l}^{*} ({\hat{s}}_{i, l})}{\partial {\hat{s}}_{i, l}}$ $Mathematical equation: $ \frac{\partial J^{*}_{i,l}(\hat{s}_{i,l})}{\partial\hat{s}_{i,l}} $$ is

$\begin{matrix} α_{i, l}^{*} = - \frac{\partial J_{i, l}^{*} ({\hat{s}}_{i, l})}{2 \partial {\hat{s}}_{i, l}} \cdot \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \alpha ^{*}_{i,l} = -\frac{\partial J^{*}_{i,l}(\hat{s}_{i,l})}{2\partial \hat{s}_{i,l}}\cdot \end{aligned} $$$ (27)

Then, the term $\frac{\partial J_{i, l}^{*} ({\hat{s}}_{i, l})}{\partial {\hat{s}}_{i, l}}$ $Mathematical equation: $ \frac{\partial J^{*}_{i,l}(\hat{s}_{i,l})}{\partial\hat{s}_{i,l}} $$ is decomposed as

$\begin{matrix} \frac{\partial J_{i, l}^{*} ({\hat{s}}_{i, l})}{\partial {\hat{s}}_{i, l}} = 2 a_{i, l} {\hat{s}}_{i, l} - \frac{2 N (τ_{i, l}) {\hat{ϑ}}_{i, l} {\hat{s}}_{i, l} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l} + ν_{i, l}^{2}}} - \frac{2 N ({\overset{˘}{τ}}_{i, l}) {\hat{ρ}}_{i, l} {\hat{s}}_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} + ν_{i, l}^{2}}} + J_{i, l}^{0} ({\bar{\hat{ζ}}}_{i, l}, {\hat{s}}_{i, l}), \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \frac{\partial J^{*}_{i,l}(\hat{s}_{i,l})}{\partial \hat{s}_{i,l}} = 2a_{i,l}\hat{s}_{i,l}-\frac{2N(\tau _{i,l})\hat{\vartheta }_{i,l} \hat{s}_{i,l}\bar{R}^{T}_{i,l}\bar{R}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l}\bar{R}^{T}_{i,l}\bar{R}_{i,l} +\nu ^{2}_{i,l}}}-\frac{2N(\breve{\tau }_{i,l})\hat{\rho }_{i,l} \hat{s}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l}+\nu ^{2}_{i,l}}} +J^{0}_{i,l}(\bar{\hat{\zeta }}_{i,l},\hat{s}_{i,l}), \end{aligned} $$$ (28)

where a_i, l is positive parameter, $N (τ_{i, l}) = exp (τ_{i, l}^{2}) sin (π τ_{i, l})$ $Mathematical equation: $ N(\tau_{i,l})= \exp(\tau_{i,l}^{2})\sin(\pi\tau_{i,l}) $$ , $N ({\overset{˘}{τ}}_{i, l}) = exp ({\overset{˘}{τ}}_{i, l}^{2}) sin (π {\overset{˘}{τ}}_{i, l})$ $Mathematical equation: $ N(\breve{\tau}_{i,l})= \exp(\breve{\tau}^{2}_{i,l})\sin(\pi\breve{\tau}_{i,l}) $$ , $ν_{i, l} = δ_{i, l} exp (- {\overset{˘}{δ}}_{i, l} t)$ $Mathematical equation: $ \nu_{i,l}=\delta_{i,l}\exp(-\breve{\delta}_{i,l}t) $$ , δ_i, l and ${\overset{˘}{δ}}_{i, l}$ $Mathematical equation: $ \breve{\delta}_{i,l} $$ are positive constants, ${\bar{R}}_{i, l}$ $Mathematical equation: $ \bar{R}_{i,l} $$ denotes the basis function vector, and $J_{i, l}^{0} ({\bar{\hat{x}}}_{i, l}, {\hat{s}}_{i, l}) = - 2 a_{i, l} {\hat{s}}_{i, l} + \frac{2 N (τ_{i, l}) {\hat{ϑ}}_{i, l} {\hat{s}}_{i, l} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l} + ν_{i, l}^{2}}} + \frac{2 N ({\overset{˘}{τ}}_{i, l}) {\hat{ρ}}_{i, l} {\hat{s}}_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} + ν_{i, l}^{2}}} + \frac{\partial J_{i, l}^{*} ({\hat{s}}_{i, l})}{\partial {\hat{s}}_{i, l}}$ $Mathematical equation: $ J^{0}_{i,l}(\bar{\hat{x}}_{i,l},\hat{s}_{i,l})=-2a_{i,l} \hat{s}_{i,l}+\frac{2N(\tau_{i,l})\hat{\vartheta}_{i,l} \hat{s}_{i,l}\bar{R}^{T}_{i,l}\bar{R}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l} \bar{R}^{T}_{i,l}\bar{R}_{i,l}+\nu^{2}_{i,l}}}+\frac{2N(\breve{\tau}_{i,l}) \hat{\rho}_{i,l}\hat{s}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l}+\nu^{2}_{i,l}}}+\frac{\partial J^{*}_{i,l}(\hat{s}_{i,l})}{\partial\hat{s}_{i,l}} $$ .

On the grounds of (28), the optimal control (27) is expressed as

$\begin{matrix} α_{i, l}^{*} = - a_{i, l} {\hat{s}}_{i, l} + \frac{N (τ_{i, l}) {\hat{ϑ}}_{i, l} {\hat{s}}_{i, l} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l} + ν_{i, l}^{2}}} + \frac{N ({\overset{˘}{τ}}_{i, l}) {\hat{ρ}}_{i, l} {\hat{s}}_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} + ν_{i, l}^{2}}} - \frac{1}{2} J_{i, l}^{0} ({\bar{\hat{ζ}}}_{i, l}, {\hat{s}}_{i, l}) . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \alpha ^{*}_{i,l}=-a_{i,l}\hat{s}_{i,l}+\frac{N(\tau _{i,l}) \hat{\vartheta }_{i,l}\hat{s}_{i,l}\bar{R}^{T}_{i,l} \bar{R}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l}\bar{R}^{T}_{i,l} \bar{R}_{i,l}+\nu ^{2}_{i,l}}}+\frac{N(\breve{\tau }_{i,l}) \hat{\rho }_{i,l}\hat{s}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l}+\nu ^{2}_{i,l}}} -\frac{1}{2}J^{0}_{i,l}(\bar{\hat{\zeta }}_{i,l},\hat{s}_{i,l}). \end{aligned} $$$

Since $J_{i, l}^{0} ({\bar{\hat{ζ}}}_{i, l}, {\hat{s}}_{i, l})$ $Mathematical equation: $ J^{0}_{i,l}(\bar{\hat{\zeta}}_{i,l},\hat{s}_{i,l}) $$ is unknown but continuous, NNs are employed to estimate $J_{i, l}^{0} ({\bar{\hat{ζ}}}_{i, l}, {\hat{s}}_{i, l})$ $Mathematical equation: $ J^{0}_{i,l}(\bar{\hat{\zeta}}_{i,l},\hat{s}_{i,l}) $$ . Then, we have

$\begin{matrix} J_{i, l}^{0} = W_{J i, l}^{* T} {\bar{R}}_{J i, l} ({\bar{\hat{ζ}}}_{i, l}, {\hat{s}}_{i, l}) + ϵ_{J i, l} ({\bar{\hat{ζ}}}_{i, l}, {\hat{s}}_{i, l}), \end{matrix}$ $Mathematical equation: $$ \begin{aligned} J^{0}_{i,l} = W^{*T}_{Ji,l}\bar{R}_{Ji,l}(\bar{\hat{\zeta }}_{i,l},\hat{s}_{i,l}) +\epsilon _{Ji,l}(\bar{\hat{\zeta }}_{i,l},\hat{s}_{i,l}), \end{aligned} $$$ (29)

where ϵ_J
i, l represents the approximation error.

According to (29), we get $\frac{\partial J_{i, l}^{*} ({\hat{s}}_{i, l})}{\partial {\hat{s}}_{i, l}}$ $Mathematical equation: $ \frac{\partial J^{*}_{i,l}(\hat{s}_{i,l})}{\partial\hat{s}_{i,l}} $$ and $α_{i, l}^{*}$ $Mathematical equation: $ \alpha^{*}_{i,l} $$ as

$\begin{matrix} \frac{\partial J_{i, l}^{*} ({\hat{s}}_{i, l})}{\partial {\hat{s}}_{i, l}} & = 2 a_{i, l} {\hat{s}}_{i, l} - \frac{2 N (τ_{i, l}) {\hat{ϑ}}_{i, l} {\hat{s}}_{i, l} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l} + ν_{i, l}^{2}}} - \frac{2 N ({\overset{˘}{τ}}_{i, l}) {\hat{ρ}}_{i, l} {\hat{s}}_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} + ν_{i, l}^{2}}} + W_{J i, l}^{* T} {\bar{R}}_{J i, l} + ϵ_{J i, l}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \frac{\partial J^{*}_{i,l}(\hat{s}_{i,l})}{\partial \hat{s}_{i,l}}&=2a_{i,l} \hat{s}_{i,l}-\frac{2N(\tau _{i,l})\hat{\vartheta }_{i,l}\hat{s}_{i,l} \bar{R}^{T}_{i,l}\bar{R}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l} \bar{R}^{T}_{i,l}\bar{R}_{i,l}+\nu ^{2}_{i,l}}} -\frac{2N(\breve{\tau }_{i,l})\hat{\rho }_{i,l}\hat{s}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l}+\nu ^{2}_{i,l}}}+W^{*T}_{Ji,l} \bar{R}_{Ji,l}+\epsilon _{Ji,l}, \end{aligned} $$$ (30)

$\begin{matrix} α_{i, l}^{*} & = - a_{i, l} {\hat{s}}_{i, l} + \frac{N (τ_{i, l}) {\hat{ϑ}}_{i, l} {\hat{s}}_{i, l} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l} + ν_{i, l}^{2}}} + \frac{N ({\overset{˘}{τ}}_{i, l}) {\hat{ρ}}_{i, l} {\hat{s}}_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} + ν_{i, l}^{2}}} - \frac{1}{2} (W_{J i, l}^{* T} {\bar{R}}_{J i, l} + ϵ_{J i, l}) . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \alpha ^{*}_{i,l}&=-a_{i,l}\hat{s}_{i,l}+\frac{N(\tau _{i,l}) \hat{\vartheta }_{i,l}\hat{s}_{i,l}\bar{R}^{T}_{i,l}\bar{R}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l}\bar{R}^{T}_{i,l}\bar{R}_{i,l}+\nu ^{2}_{i,l}}} +\frac{N(\breve{\tau }_{i,l})\hat{\rho }_{i,l}\hat{s}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l}+\nu ^{2}_{i,l}}}-\frac{1}{2} (W^{*T}_{Ji,l}\bar{R}_{Ji,l}+\epsilon _{Ji,l}). \end{aligned} $$$ (31)

The optimal control items (30) and (31) contain the unknown ideal weight $W_{J i, l}^{*}$ $Mathematical equation: $ W^{*}_{Ji,l} $$ . According to (11) and (12), $\frac{\partial {\hat{J}}_{i, l}^{*} ({\hat{s}}_{i, l})}{\partial {\hat{s}}_{i, l}}$ $Mathematical equation: $ \frac{\partial\hat{J}^{*}_{i,l}(\hat{s}_{i,l})}{\partial\hat{s}_{i,l}} $$ and ${\hat{α}}_{i, l}^{*}$ $Mathematical equation: $ \hat{\alpha}^{*}_{i,l} $$ pave the way with the critic-actor NN to tackle such intractable problem, which presented as

$\begin{matrix} \frac{\partial {\hat{J}}_{i, l}^{*} ({\hat{s}}_{i, l})}{\partial {\hat{s}}_{i, l}} & = 2 a_{i, l} {\hat{s}}_{i, l} - \frac{2 N (τ_{i, l}) {\hat{ϑ}}_{i, l} {\hat{s}}_{i, l} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l} + ν_{i, l}^{2}}} - \frac{2 N ({\overset{˘}{τ}}_{i, l}) {\hat{ρ}}_{i, l} {\hat{s}}_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} + ν_{i, l}^{2}}} + {\hat{W}}_{c i, l}^{T} (t) {\bar{R}}_{J i, l}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \frac{\partial \hat{J}^{*}_{i,l}(\hat{s}_{i,l})}{\partial \hat{s}_{i,l}}&=2a_{i,l}\hat{s}_{i,l}-\frac{2N(\tau _{i,l})\hat{\vartheta }_{i,l} \hat{s}_{i,l}\bar{R}^{T}_{i,l}\bar{R}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l} \bar{R}^{T}_{i,l}\bar{R}_{i,l}+\nu ^{2}_{i,l}}} -\frac{2N(\breve{\tau }_{i,l})\hat{\rho }_{i,l}\hat{s}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l}+\nu ^{2}_{i,l}}} +\hat{W}^{T}_{ci,l}(t)\bar{R}_{Ji,l},\end{aligned} $$$ (32)

$\begin{matrix} {\hat{α}}_{i, l}^{*} & = - a_{i, l} {\hat{s}}_{i, l} + \frac{N (τ_{i, l}) {\hat{ϑ}}_{i, l} {\hat{s}}_{i, l} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l} + ν_{i, l}^{2}}} + \frac{N ({\overset{˘}{τ}}_{i, l}) {\hat{ρ}}_{i, l} {\hat{s}}_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} + ν_{i, l}^{2}}} - \frac{1}{2} {\hat{W}}_{a i, l}^{T} (t) {\bar{R}}_{J i, l} . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \hat{\alpha }^{*}_{i,l}&=-a_{i,l}\hat{s}_{i,l} +\frac{N(\tau _{i,l})\hat{\vartheta }_{i,l}\hat{s}_{i,l} \bar{R}^{T}_{i,l}\bar{R}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l} \bar{R}^{T}_{i,l}\bar{R}_{i,l}+\nu ^{2}_{i,l}}} +\frac{N(\breve{\tau }_{i,l})\hat{\rho }_{i,l}\hat{s}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l}+\nu ^{2}_{i,l}}}-\frac{1}{2} \hat{W}^{T}_{ai,l}(t)\bar{R}_{Ji,l}. \end{aligned} $$$ (33)

Similar to the previous method, the updating laws ${\dot{\hat{W}}}_{c i, l} (t)$ $Mathematical equation: $ \dot{\hat{W}}_{ci,l}(t) $$ and ${\dot{\hat{W}}}_{a i, l} (t)$ $Mathematical equation: $ \dot{\hat{W}}_{ai,l}(t) $$ are crafted as

$\begin{matrix} \begin{matrix} {\begin{matrix} {\dot{\hat{W}}}_{c i, l} (t) & = - ν_{i, l} β_{c i, l} {\bar{R}}_{J i, l} {\bar{R}}_{J i, l}^{T} {\hat{W}}_{c i, l} (t), \\ {\dot{\hat{W}}}_{a i, l} (t) & = - {\bar{R}}_{J i, l} {\bar{R}}_{J i, l}^{T} (ν_{i, l} β_{a i, l} ({\hat{W}}_{a i, l} (t) - {\hat{W}}_{c i, l} (t)) + ν_{i, l} β_{c i, l} {\hat{W}}_{c i, l} (t)) . \end{matrix} \end{matrix} \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \begin{split} \left\{ \begin{aligned} \dot{\hat{W}}_{ci,l}(t)&=-\nu _{i,l}\beta _{ci,l} \bar{R}_{Ji,l}\bar{R}^{T}_{Ji,l}\hat{W}_{ci,l}(t),\\ \dot{\hat{W}}_{ai,l}(t)&=-\bar{R}_{Ji,l}\bar{R}^{T}_{Ji,l} (\nu _{i,l}\beta _{ai,l}\big (\hat{W}_{ai,l}(t)-\hat{W}_{ci,l} (t)\big )+\nu _{i,l}\beta _{ci,l}\hat{W}_{ci,l}(t)). \end{aligned} \right. \end{split} \end{aligned} $$$ (34)

where $β_{a i, l} > \frac{1}{2}$ $Mathematical equation: $ \beta_{ai,l} > \frac{1}{2} $$ and $β_{a i, l} > β_{c i, l} > \frac{β_{a i, l}}{2} > 0$ $Mathematical equation: $ \beta_{ai,l} > \beta_{ci,l} > \frac{\beta_{ai,l}}{2} > 0 $$ .

Establishing the Lyapunov function as

$\begin{matrix} V_{i, l} (t) = \frac{1}{2} s_{i, l}^{2} + \frac{1}{2} {\tilde{ϑ}}_{i, l}^{2} + \frac{1}{2} {\tilde{ρ}}_{i, l}^{2} + \frac{1}{2} {\tilde{W}}_{c i, l}^{T} (t) {\tilde{W}}_{c i, l} (t) + \frac{1}{2} {\tilde{W}}_{a i, l}^{T} (t) {\tilde{W}}_{a i, l} (t) + V_{i, l - 1}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} V_{i,l}(t)=\frac{1}{2}s^{2}_{i,l}+\frac{1}{2} \tilde{\vartheta }^{2}_{i,l}+\frac{1}{2}\tilde{\rho }^{2}_{i,l}+\frac{1}{2} \tilde{W}^{T}_{ci,l}(t)\tilde{W}_{ci,l}(t) +\frac{1}{2}\tilde{W}^{T}_{ai,l}(t)\tilde{W}_{ai,l}(t)+V_{i,l-1}, \end{aligned} $$$

where ${\tilde{W}}_{c i, l} (t) = {\hat{W}}_{c i, l} - W_{J i, l}^{*}$ $Mathematical equation: $ \tilde{W}_{ci,l}(t)=\hat{W}_{ci,l}-W^{*}_{Ji,l} $$ and ${\tilde{W}}_{a i, l} (t) = {\hat{W}}_{a i, l} - W_{J i, l}^{*}$ $Mathematical equation: $ \tilde{W}_{ai,l}(t)=\hat{W}_{ai,l}-W^{*}_{Ji,l} $$ . According to (6), ${\dot{V}}_{i, l} (t)$ $Mathematical equation: $ \dot{V}_{i,l}(t) $$ is derived as

$\begin{matrix} {\dot{V}}_{i, l} (t) = & s_{i, l} s_{i, l + 1} + s_{i, l} ϖ_{i, l} {\hat{α}}_{i, l}^{*} + s_{i, l} G_{i, l} ({\bar{π}}_{i, l}) + {\tilde{ϑ}}_{i, l} {\dot{\hat{ϑ}}}_{i, l} + {\tilde{ρ}}_{i, l} {\dot{\hat{ρ}}}_{i, l} + {\tilde{W}}_{c i, l}^{T} (t) {\dot{\hat{W}}}_{c i, l} (t) \\ + {\tilde{W}}_{a i, l}^{T} (t) {\dot{\hat{W}}}_{a i, l} (t) + {\dot{V}}_{i, l - 1}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{V}_{i,l}(t) =&s_{i,l}s_{i,l+1}+s_{i,l}\varpi _{i,l}\hat{\alpha }^{*}_{i,l} +s_{i,l}G_{i,l}(\bar{\pi }_{i,l})+\tilde{\vartheta }_{i,l} \dot{\hat{\vartheta }}_{i,l} +\tilde{\rho }_{i,l}\dot{\hat{\rho }}_{i,l} +\tilde{W}^{T}_{ci,l}(t)\dot{\hat{W}}_{ci,l}(t)\nonumber \\&+\tilde{W}^{T}_{ai,l}(t) \dot{\hat{W}}_{ai,l}(t)+\dot{V}_{i,l-1}, \end{aligned} $$$ (35)

where $G_{i, l} ({\bar{π}}_{i, l}) = g_{i, l} - {\dot{ϖ}}_{i, l} {\hat{α}}_{i, l - 1}^{*} - ϖ_{i, l} {\dot{\hat{α}}}_{i, l - 1}^{*}$ $Mathematical equation: $ G_{i,l}(\bar{\pi}_{i,l}) = g_{i,l}-\dot{\varpi}_{i,l}\hat{\alpha}^{*}_{i,l-1} - \varpi_{i,l}\dot{\hat{\alpha}}^{*}_{i,l-1} $$ , ${\bar{π}}_{i, l} = {[{\bar{\hat{ζ}}}_{i, l}, y_{r}, {\bar{y}}_{r}^{(l)}]}^{T}$ $Mathematical equation: $ \bar{\pi}_{i,l} = [\bar{\hat{\zeta}}_{i,l}, y_{r}, \bar{y}^{(l)}_{r}]^{T} $$ , and ${\bar{y}}_{r}^{(l)} = {[{\dot{y}}_{r}, \dots, y_{r}^{(l)}]}^{T}$ $Mathematical equation: $ \bar{y}^{(l)}_{r}=[\dot{y}_{r}, \ldots, y^{(l)}_{r}]^{T} $$ .

Due to the fact that $G_{i, l} ({\bar{π}}_{i, l})$ $Mathematical equation: $ G_{i,l}(\bar{\pi}_{i,l}) $$ includes unknown functions, the RBF NNs $ξ_{i, l}^{* T} {\bar{R}}_{i, l} ({\bar{π}}_{i, l})$ $Mathematical equation: $ \xi^{\ast T}_{i,l}\bar{R}_{i,l}(\bar{\pi}_{i,l}) $$ are used to estimate $G_{i, l} ({\bar{π}}_{i, l})$ $Mathematical equation: $ G_{i,l}(\bar{\pi}_{i,l}) $$ . Then, one has

$\begin{matrix} s_{i, l} G_{i, l} ({\bar{π}}_{i, l}) = s_{i, l} (ξ_{i, l}^{* T} {\bar{R}}_{i, l} ({\bar{π}}_{i, l}) + ϵ_{i, l} ({\bar{π}}_{i, l})) \leq \frac{ϑ_{i, l} {\hat{s}}_{i, l}^{2} {\bar{R}}_{i, l}^{T} R_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l} + ν_{i, l}^{2}}} + θ_{i, l} ν_{i, l} + s_{i, l} ϵ_{i, l}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} s_{i,l}G_{i,l}(\bar{\pi }_{i,l})=s_{i,l}\Big (\xi _{i,l}^{*T}\bar{R}_{i,l}(\bar{\pi }_{i,l})+\epsilon _{i,l}(\bar{\pi }_{i,l})\Big ) \le \frac{\vartheta _{i,l}\hat{s}^{2}_{i,l}\bar{R}^{T}_{i,l}R_{i,l}}{\sqrt{\hat{s}^{2}_{i,l}\bar{R}^{T}_{i,l} \bar{R}_{i,l}+\nu ^{2}_{i,l}}}+\theta _{i,l}\nu _{i,l}+ s_{i,l} \epsilon _{i,l}, \end{aligned} $$$ (36)

where $| ϵ_{i, l} ({\bar{π}}_{i, l}) | \leq ε_{i, l}$ $Mathematical equation: $ \vert \epsilon_{i,l}(\bar{\pi}_{i,l}) \vert\leq\varepsilon_{i,l} $$ .

By substituting (32), (33) and (36) into (35), it can obtains that

$\begin{matrix} {\dot{V}}_{i, l} (t) \leq & s_{i, l} s_{i, l + 1} + s_{i, l} ϖ_{i, l} (- a_{i, l} {\hat{s}}_{i, l} - \frac{1}{2} {\hat{W}}_{a i, l}^{T} (t) {\bar{R}}_{J i, l} + \frac{N (τ_{i, l}) {\hat{ϑ}}_{i, l} {\hat{s}}_{i, l} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l} + ν_{i, l}^{2}}} + \frac{N ({\overset{˘}{τ}}_{i, l}) {\hat{ρ}}_{i, l} {\hat{s}}_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} + ν_{i, l}^{2}}}) \\ + \frac{ϑ_{i, l} {\hat{s}}_{i, l}^{2} {\bar{R}}_{i, l}^{T} R_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l} + ν_{i, l}^{2}}} + s_{i, l} ϵ_{i, l} + ϑ_{i, l} ν_{i, l} + {\tilde{ϑ}}_{i, l} {\dot{\hat{ϑ}}}_{i, l} + {\tilde{ρ}}_{i, l} {\dot{\hat{ρ}}}_{i, l} + {\tilde{W}}_{c i, l}^{T} (t) {\dot{\hat{W}}}_{c i, l} (t) \\ + {\tilde{W}}_{a i, l}^{T} (t) {\dot{\hat{W}}}_{a i, l} (t) + {\dot{V}}_{i, l - 1} . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{V}_{i,l}(t)\le &s_{i,l}s_{i,l+1}+s_{i,l}\varpi _{i,l} \left(-a_{i,l}\hat{s}_{i,l}-\frac{1}{2}\hat{W}^{T}_{ai,l}(t) \bar{R}_{Ji,l}+\frac{N(\tau _{i,l})\hat{\vartheta }_{i,l} \hat{s}_{i,l}\bar{R}^{T}_{i,l}\bar{R}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l}\bar{R}^{T}_{i,l} \bar{R}_{i,l}+\nu ^{2}_{i,l}}}+\frac{N(\breve{\tau }_{i,l}) \hat{\rho }_{i,l}\hat{s}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l}+\nu ^{2}_{i,l}}}\right)\nonumber \\&+\frac{\vartheta _{i,l}\hat{s}^{2}_{i,l}\bar{R}^{T}_{i,l}R_{i,l}}{\sqrt{\hat{s}^{2}_{i,l}\bar{R}^{T}_{i,l}\bar{R}_{i,l}+\nu ^{2}_{i,l}}}+s_{i,l} \epsilon _{i,l}+\vartheta _{i,l}\nu _{i,l}+\tilde{\vartheta }_{i,l} \dot{\hat{\vartheta }}_{i,l}+\tilde{\rho }_{i,l}\dot{\hat{\rho }}_{i,l}+\tilde{W}^{T}_{ci,l}(t) \dot{\hat{W}}_{ci,l}(t)\nonumber \\&+\tilde{W}^{T}_{ai,l}(t)\dot{\hat{W}}_{ai,l}(t)+\dot{V}_{i,l-1}. \end{aligned} $$$ (37)

As in the first step of (20), we get

$\begin{matrix} \begin{matrix} {\begin{matrix} s_{i, l} s_{i, l + 1} \leq \frac{1}{2} s_{i, l}^{2} + \frac{1}{2} s_{i, l + 1}^{2}, \\ | s_{i, l} | (ε_{i, l} + | \frac{1}{2} ϖ_{i, l} {\hat{W}}_{a i, l}^{T} (t) {\bar{R}}_{J i, l} |) \leq \frac{ρ_{i, l} {\hat{s}}_{i, l}^{2}}{\sqrt{{\hat{s}}_{i, l}^{2} + ν_{i, l}^{2}}} + ρ_{i, l} ν_{i, l} . \end{matrix} \end{matrix} \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \begin{split} \left\{ \begin{aligned}&s_{i,l}s_{i,l+1}\le \frac{1}{2}s^{2}_{i,l}+\frac{1}{2}s^{2}_{i,l+1},\\&\vert s_{i,l}\vert \Big (\varepsilon _{i,l}+\vert \frac{1}{2} \varpi _{i,l}\hat{W}^{T}_{ai,l}(t)\bar{R}_{Ji,l} \vert \Big )\le \frac{\rho _{i,l}\hat{s}^{2}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l} +\nu ^{2}_{i,l}}}+\rho _{i,l}\nu _{i,l}. \end{aligned} \right. \end{split} \end{aligned} $$$ (38)

The adaptive laws can be designed as

$\begin{matrix} {\dot{\hat{ϑ}}}_{i, l} = \frac{{\hat{s}}_{i, l}^{2} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l} + ν_{i, l}^{2}}} - ν_{i, l} {\hat{ϑ}}_{i, l}, {\dot{\hat{ρ}}}_{i, l} = \frac{{\hat{s}}_{i, l}^{2}}{\sqrt{{\hat{s}}_{i, l}^{2} + ν_{i, l}^{2}}} - ν_{i, l} {\hat{ρ}}_{i, l} . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{\hat{\vartheta }}_{i,l}=\frac{\hat{s}^{2}_{i,l}\bar{R}^{T}_{i,l} \bar{R}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l}\bar{R}^{T}_{i,l} \bar{R}_{i,l}+\nu ^{2}_{i,l}}}-\nu _{i,l}\hat{\vartheta }_{i,l}, \qquad \dot{\hat{\rho }}_{i,l}=\frac{\hat{s}^{2}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l} +\nu ^{2}_{i,l}}}-\nu _{i,l}\hat{\rho }_{i,l}. \end{aligned} $$$ (39)

By substituting (34), (38) and (39) into (37), we can yield that

$\begin{matrix} {\dot{V}}_{i, l} (t) \leq & \frac{1}{2} s_{i, l + 1}^{2} - (a_{i, l} - \frac{1}{2}) s_{i, l}^{2} + (N (τ_{i, l}) ϖ_{i, l}^{2} + 1) {\dot{τ}}_{i, l} + (N ({\overset{˘}{τ}}_{i, l}) ϖ_{i, l}^{2} + 1) {\dot{\overset{˘}{τ}}}_{i, l} \\ - {\tilde{W}}_{c i, l}^{T} (t) ν_{i, l} β_{c i, l} {\bar{R}}_{J i, l}^{T} {\bar{R}}_{J i, l} {\hat{W}}_{c i, l} (t) - {\tilde{W}}_{a i, l}^{T} (t) {\bar{R}}_{J i, l}^{T} {\bar{R}}_{J i, l} (ν_{i, l} β_{a i, l} ({\hat{W}}_{a i, l} (t) \\ - {\hat{W}}_{c i, l} (t)) + ν_{i, l} β_{c i, l} {\hat{W}}_{c i, l} (t)) - ν_{i, l} {\tilde{ϑ}}_{i, l} {\hat{ϑ}}_{i, l} - ν_{i, l} {\tilde{ρ}}_{i, l} {\hat{ρ}}_{i, l} + ϑ_{i, l} ν_{i, l} + ρ_{i, l} ν_{i, l} + {\dot{V}}_{i, l - 1}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{V}_{i,l}(t)\le &\frac{1}{2}s^{2}_{i,l+1}-\Big (a_{i,l} -\frac{1}{2}\Big )s^{2}_{i,l}+\big (N(\tau _{i,l})\varpi _{i,l}^{2}+1\big ) \dot{\tau }_{i,l}+\big (N(\breve{\tau }_{i,l})\varpi _{i,l}^{2}+1\big ) \dot{\breve{\tau }}_{i,l}\nonumber \\&-\tilde{W}^{T}_{ci,l}(t)\nu _{i,l}\beta _{ci,l}\bar{R}^{T}_{Ji,l} \bar{R}_{Ji,l}\hat{W}_{ci,l}(t)-\tilde{W}^{T}_{ai,l}(t)\bar{R}^{T}_{Ji,l} \bar{R}_{Ji,l}\Big (\nu _{i,l}\beta _{ai,l}\big (\hat{W}_{ai,l}(t)\nonumber \\&-\hat{W}_{ci,l}(t)\big )+\nu _{i,l}\beta _{ci,l}\hat{W}_{ci,l}(t)\Big ) -\nu _{i,l}\tilde{\vartheta }_{i,l}\hat{\vartheta }_{i,l} -\nu _{i,l}\tilde{\rho }_{i,l}\hat{\rho }_{i,l}+\vartheta _{i,l}\nu _{i,l} +\rho _{i,l}\nu _{i,l}+\dot{V}_{i,l-1}, \end{aligned} $$$ (40)

where ${\dot{τ}}_{i, l} = \frac{{\hat{θ}}_{i, l} {\hat{s}}_{i, l}^{2} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l}}{\sqrt{{\hat{s}}_{i, l}^{2} {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l} + ν_{i, l}^{2}}}$ $Mathematical equation: $ \dot{\tau}_{i,l} = \frac{\hat{\theta}_{i,l}\hat{s}^{2}_{i,l} \bar{R}^{T}_{i,l}\bar{R}_{i,l}}{\sqrt{\hat{s}^{2}_{i,l} \bar{R}^{T}_{i,l}\bar{R}_{i,l}+\nu^{2}_{i,l}}} $$ and ${\dot{\overset{˘}{τ}}}_{i, l} = \frac{{\hat{ρ}}_{i, l} {\hat{s}}_{i, l}^{2}}{\sqrt{{\hat{s}}_{i, l}^{2} + ν_{i, l}^{2}}}$ $Mathematical equation: $ \dot{\breve{\tau}}_{i,l} = \frac{\hat{\rho}_{i,l}\hat{s}^{2}_{i,l}} {\sqrt{\hat{s}^{2}_{i,l}+\nu^{2}_{i,l}}} $$ .

By using ${\tilde{W}}_{c i, l} (t) = {\hat{W}}_{c i, l} - W_{J i, l}^{*}$ $Mathematical equation: $ \tilde{W}_{ci,l}(t)=\hat{W}_{ci,l}-W^{*}_{Ji,l} $$ and ${\tilde{W}}_{a i, l} (t) = {\hat{W}}_{a i, l} - W_{J i, l}^{*}$ $Mathematical equation: $ \tilde{W}_{ai,l}(t)=\hat{W}_{ai,l}-W^{*}_{Ji,l} $$ , one obtains

$\begin{matrix} {\tilde{W}}_{c i, l}^{T} (t) {\bar{R}}_{J i, l}^{T} {\bar{R}}_{J i, l} {\hat{W}}_{c i, l} (t) = & \frac{1}{2} {\hat{W}}_{c i, l}^{T} (t) {\bar{R}}_{J i, l}^{T} {\bar{R}}_{J i, l} {\hat{W}}_{c i, l} (t) + \frac{1}{2} {\tilde{W}}_{c i, l}^{T} (t) {\bar{R}}_{J i, l} {\bar{R}}_{J i, l}^{T} {\tilde{W}}_{c i, l} (t) \\ - \frac{1}{2} (W_{J i, l}^{* T} {\bar{R}}_{J i, l})^{2}, \\ {\tilde{W}}_{a i, l}^{T} (t) {\bar{R}}_{J i, l}^{T} {\bar{R}}_{J i, l} {\hat{W}}_{a i, l} (t) = & \frac{1}{2} {\hat{W}}_{a i, l}^{T} (t) {\bar{R}}_{J i, l}^{T} {\bar{R}}_{J i, l} {\hat{W}}_{a i, l} (t) + \frac{1}{2} {\tilde{W}}_{a i, l}^{T} (t) {\bar{R}}_{i, l}^{T} {\bar{R}}_{i, l} {\tilde{W}}_{a i, l} (t) \\ - \frac{1}{2} (W_{J i, l}^{* T} {\bar{R}}_{J i, l})^{2} . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \tilde{W}^{T}_{ci,l}(t)\bar{R}^{T}_{Ji,l}\bar{R}_{Ji,l}\hat{W}_{ci,l}(t) =&\frac{1}{2}\hat{W}^{T}_{ci,l}(t)\bar{R}^{T}_{Ji,l}\bar{R}_{Ji,l} \hat{W}_{ci,l}(t)+\frac{1}{2}\tilde{W}^{T}_{ci,l}(t)\bar{R}_{Ji,l} \bar{R}^{T}_{Ji,l}\tilde{W}_{ci,l}(t)\nonumber \\&-\frac{1}{2}\big (W^{*T}_{Ji,l}\bar{R}_{Ji,l}\big )^{2}, \\ \tilde{W}^{T}_{ai,l}(t)\bar{R}^{T}_{Ji,l}\bar{R}_{Ji,l}\hat{W}_{ai,l}(t) =&\frac{1}{2}\hat{W}^{T}_{ai,l}(t)\bar{R}^{T}_{Ji,l}\bar{R}_{Ji,l} \hat{W}_{ai,l}(t)+\frac{1}{2}\tilde{W}^{T}_{ai,l}(t)\bar{R}^{T}_{i,l} \bar{R}_{i,l}\tilde{W}_{ai,l}(t)\nonumber \\&-\frac{1}{2}\big (W^{*T}_{Ji,l}\bar{R}_{Ji,l}\big )^{2}. \end{aligned} $$$ (41) (42)

From (26) and (40)–(42), we have

$\begin{matrix} {\dot{V}}_{i, l} (t) \leq & \frac{1}{2} s_{i, l + 1}^{2} - \sum_{k = 1}^{l} ((a_{i, k} - 1) s_{i, k}^{2} + (\frac{ν_{i, k}}{2} {\tilde{ϑ}}_{i, k}^{2} + \frac{ν_{i, k} β_{c i, k}}{2} {\tilde{W}}_{c i, k}^{T} (t) {\bar{R}}_{J i, k}^{T} {\bar{R}}_{J i, k} {\tilde{W}}_{c i, k} (t) \\ + \frac{ν_{i, k}}{2} {\tilde{ρ}}_{i, k}^{2})) + \sum_{k = 1}^{l} ((N (τ_{i, k}) ϖ_{i, k}^{2} + 1) {\dot{τ}}_{i, k} + \frac{ν_{i, k} β_{c i, k}}{2} {\tilde{W}}_{a i, k}^{T} (t) {\bar{R}}_{J i, k}^{T} {\bar{R}}_{J i, k} {\tilde{W}}_{a i, k} (t) \\ + (N ({\overset{˘}{τ}}_{i, k}) ϖ_{i, k}^{2} + 1) {\dot{\overset{˘}{τ}}}_{i, k}) + \sum_{k = 1}^{l} μ_{i, k}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{V}_{i,l}(t)\le &\frac{1}{2}s^{2}_{i,l+1}-\sum ^{l}_{k=1} \Bigg (\big (a_{i,k}-1\big )s^{2}_{i,k}+\Big (\frac{\nu _{i,k}}{2} \tilde{\vartheta }^{2}_{i,k}+\frac{\nu _{i,k}\beta _{ci,k}}{2} \tilde{W}^{T}_{ci,k}(t)\bar{R}^{T}_{Ji,k} \bar{R}_{Ji,k}\tilde{W}_{ci,k}(t)\nonumber \\&+\frac{\nu _{i,k}}{2}\tilde{\rho }^{2}_{i,k}\Big )\Bigg ) +\sum ^{l}_{k=1}\Big (\big (N(\tau _{i,k})\varpi _{i,k}^{2}+1\big ) \dot{\tau }_{i,k}+\frac{\nu _{i,k}\beta _{ci,k}}{2} \tilde{W}^{T}_{ai,k}(t)\bar{R}^{T}_{Ji,k} \bar{R}_{Ji,k}\tilde{W}_{ai,k}(t) \nonumber \\&+\big (N(\breve{\tau }_{i,k})\varpi _{i,k}^{2}+1\big ) \dot{\breve{\tau }}_{i,k}\Big )+\sum ^{l}_{k=1}\mu _{i,k}, \end{aligned} $$$ (43)

where $μ_{i, k} = \frac{ν_{i, k}}{2} ϑ_{i, k}^{2} + \frac{ν_{i, k}}{2} ρ_{i, k}^{2} + (\frac{ν_{i, k} β_{a i, k}}{2} + \frac{ν_{i, k} β_{c i, k}}{2}) (W_{J i, k}^{* T} {\bar{R}}_{J i, k})^{2} + θ_{i, k} ν_{i, k} + ρ_{i, k} ν_{i, k}$ $Mathematical equation: $ \mu_{i,k} = \frac{\nu_{i,k}}{2}\vartheta^{2}_{i,k} + \frac{\nu_{i,k}}{2}\rho^{2}_{i,k}+\big(\frac{\nu_{i,k}\beta_{ai,k}}{2} + \frac{\nu_{i,k}\beta_{ci,k}}{2}\big) \big(W^{*T}_{Ji,k}\bar{R}_{Ji,k}\big)^{2} + \theta_{i,k}\nu_{i,k}+\rho_{i,k}\nu_{i,k} $$ .

Step m : The functional derivatives of ${\hat{s}}_{i, m}$ $Mathematical equation: $ \hat{s}_{i,m} $$ is calculated by the definition of (5), one has

$\begin{matrix} {\dot{\hat{s}}}_{i, m} = \frac{u_{i}}{ϖ_{i, m}} + \frac{g_{i, m}}{ϖ_{i, m}} - \frac{{\dot{ϖ}}_{i, m} {\hat{x}}_{i, m}}{ϖ_{i, m}} - {\dot{\hat{α}}}_{i, l - 1}^{*} . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{\hat{s}}_{i,m} = \frac{u_{i}}{\varpi _{i,m}} + \frac{g_{i,m}}{\varpi _{i,m}}-\frac{\dot{\varpi }_{i,m} \hat{x}_{i,m}}{\varpi _{i,m}}-\dot{\hat{\alpha }}^{*}_{i,l-1}. \end{aligned} $$$

We define $u_{i}^{*}$ $Mathematical equation: $ u^{*}_{i} $$ as the optimal controller. Comparable to step l, one has

$\begin{matrix} J_{i, m}^{*} ({\hat{s}}_{i, m}) = min_{g_{i} \in Ψ (Ω)} (\int_{t}^{\infty} K_{i, m} ({\hat{s}}_{i, m} (z), u_{i} ({\hat{s}}_{i, m})) d z) = \int_{t}^{\infty} K_{i, m} ({\hat{s}}_{i, m} (z), u_{i}^{*} ({\hat{s}}_{i, m})) d z, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} J^{*}_{i,m}(\hat{s}_{i,m}) = \min _{g_{i}\in \mathrm \Psi (\mathrm \Omega )} \Big (\int ^{\infty }_{t}K_{i,m}\big (\hat{s}_{i,m}(z),u_{i} (\hat{s}_{i,m})\big )dz\Big ) = \int ^{\infty }_{t}K_{i,m} \big (\hat{s}_{i,m}(z),u^{*}_{i}(\hat{s}_{i,m})\big )dz, \end{aligned} $$$

where $K_{i, m} ({\hat{s}}_{i, m}, u_{i}) = {\hat{s}}_{i, m}^{2} + u_{i}^{2}$ $Mathematical equation: $ K_{i,m}(\hat{s}_{i,m},u_{i}) = \hat{s}^{2}_{i,m}+u_{i}^{2} $$ is the cost function.

Under the action of the optimal intermediate controller the HJB equation is deduced as

$\begin{matrix} H_{i, m} ({\hat{s}}_{i, m}, u_{i}^{*}, \frac{\partial J_{i, m}^{*}}{\partial {\hat{s}}_{i, m}}) & = {\hat{s}}_{i, m}^{2} + u_{i}^{* 2} + \frac{\partial J_{i}^{*} ({\hat{s}}_{i, m})}{\partial {\hat{s}}_{i, m}} (\frac{u_{i}^{*}}{ϖ_{i, m}} + \frac{f_{i, m}}{ϖ_{i, m}} - \frac{{\dot{ϖ}}_{i, m} {\hat{ζ}}_{i, m}}{ϖ_{i, m}} - {\dot{\hat{α}}}_{i, m - 1}^{*}) \\ = 0 . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} H_{i,m}(\hat{s}_{i,m},u^{*}_{i}, \frac{\partial J^{*}_{i,m}}{\partial \hat{s}_{i,m}})&=\hat{s}^{2}_{i,m}+u^{*2}_{i}+\frac{\partial J^{*}_{i} (\hat{s}_{i,m})}{\partial \hat{s}_{i,m}}\Big (\frac{u^{*}_{i}}{\varpi _{i,m}}+\frac{f_{i,m}}{\varpi _{i,m}} -\frac{\dot{\varpi }_{i,m}\hat{\zeta }_{i,m}}{\varpi _{i,m}} -\dot{\hat{\alpha }}^{*}_{i,m-1}\Big ) \\&=0. \end{aligned} $$$

By calculating $\frac{\partial H_{i, m}}{\partial u_{i}^{*}} = 0$ $Mathematical equation: $ \frac{\partial H_{i,m}}{\partial u^{*}_{i}}=0 $$ , we have $u_{i}^{*} = - \frac{\partial J_{i, m}^{*} ({\hat{s}}_{i, m})}{2 ϖ_{i, m} \partial {\hat{s}}_{i, m}}$ $Mathematical equation: $ u^{*}_{i}=-\frac{\partial J^{*}_{i,m}(\hat{s}_{i,m})}{2\varpi_{i,m}\partial\hat{s}_{i,m}} $$ . According to the above analysis, the gradient term $\frac{\partial J_{i, m}^{*} ({\hat{s}}_{i, m})}{\partial {\hat{s}}_{i, m}}$ $Mathematical equation: $ \frac{\partial J^{*}_{i,m}(\hat{s}_{i,m})}{\partial\hat{s}_{i,m}} $$ is segmented as

$\begin{matrix} \frac{\partial J_{i, m}^{*} ({\hat{s}}_{i, m})}{\partial {\hat{s}}_{i, m}} = 2 ϖ_{i, m} (- α_{i, m + 1} + \frac{1}{2} J_{i, m}^{0} ({\bar{\hat{x}}}_{i, m}, {\hat{s}}_{i, m})), \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \frac{\partial J^{*}_{i,m}(\hat{s}_{i,m})}{\partial \hat{s}_{i,m}} = 2\varpi _{i,m}\Big (-\alpha _{i,m+1} + \frac{1}{2}J^{0}_{i,m}(\bar{\hat{x}}_{i,m},\hat{s}_{i,m})\Big ), \end{aligned} $$$

where $J_{i, m}^{0} = 2 α_{i, m + 1} + \frac{\partial J_{i, m}^{*} ({\hat{s}}_{i, m})}{ϖ_{i, m} \partial {\hat{s}}_{i, m}}$ $Mathematical equation: $ J^{0}_{i,m} = 2\alpha_{i,m+1}+\frac{\partial J^{*}_{i,m}(\hat{s}_{i,m})}{\varpi_{i,m}\partial\hat{s}_{i,m}} $$ .

The optimal intermediate controller $u_{i}^{*}$ $Mathematical equation: $ u^{*}_{i} $$ is given by

$\begin{matrix} u_{i}^{*} = α_{i, m + 1} - \frac{1}{2} J_{i, m}^{0}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} u^{*}_{i}=\alpha _{i,m+1}-\frac{1}{2}J^{0}_{i,m}, \end{aligned} $$$ (44)

Since $J_{i, m}^{0}$ $Mathematical equation: $ J^{0}_{i,m} $$ is unknown but continuous. With the approximation properties of NNs, we get

$\begin{matrix} J_{i, m}^{0} = W_{J i, m}^{* T} {\bar{R}}_{J i, m} ({\bar{\hat{ζ}}}_{i, m}, {\hat{s}}_{i, m}) + ϵ_{J i, m} ({\bar{\hat{ζ}}}_{i, m}, {\hat{s}}_{i, m}), \end{matrix}$ $Mathematical equation: $$ \begin{aligned} J^{0}_{i,m} = W^{*T}_{Ji,m}\bar{R}_{Ji,m} (\bar{\hat{\zeta }}_{i,m},\hat{s}_{i,m})+\epsilon _{Ji,m} (\bar{\hat{\zeta }}_{i,m},\hat{s}_{i,m}), \end{aligned} $$$ (45)

where ϵ_J
i, m represents the approximation error.

In light of (45), it follows that

$\begin{matrix} \frac{\partial J_{i, m}^{*} ({\hat{s}}_{i, m})}{\partial {\hat{s}}_{i, m}} & = 2 ϖ_{i, m} (- α_{i, m + 1} + \frac{1}{2} (W_{J i, m}^{* T} {\bar{R}}_{J i, m} + ϵ_{J i, m})), \\ u_{i}^{*} & = α_{i, m + 1} - \frac{1}{2} (W_{J i, m}^{* T} {\bar{R}}_{J i, m} + ϵ_{J i, m}) . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \frac{\partial J^{*}_{i,m}(\hat{s}_{i,m})}{\partial \hat{s}_{i,m}}&=2\varpi _{i,m}\Big (-\alpha _{i,m+1}+\frac{1}{2}\big (W^{*T}_{Ji,m} \bar{R}_{Ji,m}+\epsilon _{Ji,m}\big )\Big ), \\ u^{*}_{i}&=\alpha _{i,m+1}-\frac{1}{2}\big (W^{*T}_{Ji,m} \bar{R}_{Ji,m}+\epsilon _{Ji,m}\big ). \end{aligned} $$$

Note that $\frac{\partial J_{i, m}^{*} ({\hat{s}}_{i, m})}{\partial {\hat{s}}_{i, m}}$ $Mathematical equation: $ \frac{\partial J^{*}_{i,m}(\hat{s}_{i,m})}{\partial\hat{s}_{i,m}} $$ and $u_{i}^{*}$ $Mathematical equation: $ u^{*}_{i} $$ are unavailable because the ideal weight $W_{J i, m}^{*}$ $Mathematical equation: $ W^{*}_{Ji,m} $$ is unknown. The NNs-based RL pave the way to tackle such intractable problem.

By introducing NNs, we can obtain

$\begin{matrix} \frac{\partial {\hat{J}}_{i, m}^{*} ({\hat{s}}_{i, m})}{\partial {\hat{s}}_{i, m}} & = - 2 ϖ_{i, m} α_{i, m + 1} + ϖ_{i, m} {\hat{W}}_{c i, m}^{T} (t) {\bar{R}}_{J i, m}, \\ {\hat{u}}_{i}^{*} & = α_{i, m + 1} - \frac{1}{2} {\hat{W}}_{a i, m}^{T} (t) {\bar{R}}_{J i, m} . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \frac{\partial \hat{J}^{*}_{i,m}(\hat{s}_{i,m})}{\partial \hat{s}_{i,m}}&= -2\varpi _{i,m}\alpha _{i,m+1}+{\varpi _{i,m}} \hat{W}^{T}_{ci,m}(t)\bar{R}_{Ji,m}, \nonumber \\ \hat{u}^{*}_{i}&=\alpha _{i,m+1} -\frac{1}{2}\hat{W}^{T}_{ai,m}(t)\bar{R}_{Ji,m}. \end{aligned} $$$ (46)

Following the same procedure as before, ${\dot{\hat{W}}}_{c i, m} (t)$ $Mathematical equation: $ \dot{\hat{W}}_{ci,m}(t) $$ and ${\dot{\hat{W}}}_{a i, m} (t)$ $Mathematical equation: $ \dot{\hat{W}}_{ai,m}(t) $$ can be given by

$\begin{matrix} \begin{matrix} {\begin{matrix} {\dot{\hat{W}}}_{c i, m} (t) = & - ν_{i, m} β_{c i, m} {\bar{R}}_{J i, m} {\bar{R}}_{J i, m}^{T} {\hat{W}}_{c i, m} (t), \\ {\dot{\hat{W}}}_{a i, m} (t) = & - {\bar{R}}_{J i, m} {\bar{R}}_{J i, m}^{T} (ν_{i, m} β_{a i, m} ({\hat{W}}_{a i, m} (t) - {\hat{W}}_{c i, m} (t)) + ν_{i, m} β_{c i, m} {\hat{W}}_{c i, m} (t)) . \end{matrix} \end{matrix} \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \begin{split} \left\{ \begin{aligned} \dot{\hat{W}}_{ci,m}(t)=&-\nu _{i,m}\beta _{ci,m}\bar{R}_{Ji,m} \bar{R}^{T}_{Ji,m}\hat{W}_{ci,m}(t), \\ \dot{\hat{W}}_{ai,m}(t)=&-\bar{R}_{Ji,m}\bar{R}^{T}_{Ji,m} \Big (\nu _{i,m}\beta _{ai,m}\big (\hat{W}_{ai,m}(t) -\hat{W}_{ci,m}(t)\big )+\nu _{i,m}\beta _{ci,m}\hat{W}_{ci,m}(t)\Big ). \end{aligned} \right. \end{split} \end{aligned} $$$ (47)

where $β_{a i, m} > \frac{1}{2}$ $Mathematical equation: $ \beta_{ai,m} > \frac{1}{2} $$ and $β_{a i, m} > β_{c i, m} > \frac{β_{a i, m}}{2} > 0$ $Mathematical equation: $ \beta_{ai,m} > \beta_{ci,m} > \frac{\beta_{ai,m}}{2} > 0 $$ .

Combining ${\dot{s}}_{i, m}$ $Mathematical equation: $ \dot s_{i,m} $$ with (1) and (6), it yields

$\begin{matrix} {\dot{s}}_{i, m} = u_{i} + g_{i, m} - {\dot{ϖ}}_{i, m} {\hat{α}}_{i, m - 1}^{*} - ϖ_{i, m} {\dot{\hat{α}}}_{i, m - 1}^{*} . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{s}_{i,m} = u_{i}+g_{i,m}-\dot{\varpi }_{i,m}\hat{\alpha }^{*}_{i,m-1} -\varpi _{i,m}\dot{\hat{\alpha }}^{*}_{i,m-1}. \end{aligned} $$$

Consider the V_i, m(t) as

$\begin{matrix} V_{i, m} (t) = V_{i, m - 1} + \frac{1}{2} s_{i, m}^{2} + \frac{1}{2} {\tilde{ϑ}}_{i, m}^{2} + \frac{1}{2} {\tilde{ϑ}}_{i, m + 1}^{2} + \frac{1}{2} {\tilde{ρ}}_{i, m}^{2} + \frac{1}{2} {\tilde{W}}_{c i, m}^{T} (t) {\tilde{W}}_{c i, m} (t) + \frac{1}{2} {\tilde{W}}_{a i, m}^{T} (t) {\tilde{W}}_{a i, m} (t), \end{matrix}$ $Mathematical equation: $$ \begin{aligned} V_{i,m}(t) = V_{i,m-1}+\frac{1}{2}s^{2}_{i,m}+\frac{1}{2} \tilde{\vartheta }^{2}_{i,m}+\frac{1}{2} \tilde{\vartheta }^{2}_{i,m+1}+\frac{1}{2}\tilde{\rho }^{2}_{i,m} +\frac{1}{2}\tilde{W}^{T}_{ci,m}(t)\tilde{W}_{ci,m}(t) +\frac{1}{2}\tilde{W}^{T}_{ai,m}(t)\tilde{W}_{ai,m}(t), \end{aligned} $$$ (48)

where ${\tilde{W}}_{c i, m} (t) = {\hat{W}}_{c i, m} - W_{J i, m}^{*}$ $Mathematical equation: $ \tilde{W}_{ci,m}(t)=\hat{W}_{ci,m}-W^{*}_{Ji,m} $$ and ${\tilde{W}}_{a i, m} (t) = {\hat{W}}_{a i, m} - W_{J i, m}^{*}$ $Mathematical equation: $ \tilde{W}_{ai,m}(t)=\hat{W}_{ai,m}-W^{*}_{Ji,m} $$ . Then, according with (48) yields that

$\begin{matrix} {\dot{V}}_{i, m} (t) = & s_{i, m} G_{i, m} ({\bar{π}}_{i, m}) + {\tilde{ϑ}}_{i, m} {\dot{\hat{ϑ}}}_{i, m} + {\tilde{ϑ}}_{i, m + 1} {\dot{\hat{ϑ}}}_{i, m + 1} + {\tilde{ρ}}_{i, m} {\dot{\hat{ρ}}}_{i, m} + {\tilde{W}}_{c i, m}^{T} {\dot{\hat{W}}}_{c i, m} (t) \\ + {\tilde{W}}_{a i, m}^{T} {\dot{\hat{W}}}_{a i, m} + s_{i, m} u_{i} + {\dot{V}}_{i, m - 1}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{V}_{i,m}(t)=&s_{i,m}G_{i,m}(\bar{\pi }_{i,m}) +\tilde{\vartheta }_{i,m}\dot{\hat{\vartheta }}_{i,m} +\tilde{\vartheta }_{i,m+1} \dot{\hat{\vartheta }}_{i,m+1}+\tilde{\rho }_{i,m} \dot{\hat{\rho }}_{i,m}+\tilde{W}^{T}_{ci,m} \dot{\hat{W}}_{ci,m}(t)\nonumber \\&+\tilde{W}^{T}_{ai,m} \dot{\hat{W}}_{ai,m}+s_{i,m}u_{i}+\dot{V}_{i,m-1}, \end{aligned} $$$ (49)

where $G_{i, m} ({\bar{π}}_{i, m}) = g_{i, m} - {\dot{ϖ}}_{i, m} {\hat{α}}_{i, m - 1}^{*} - ϖ_{i, m} {\dot{\hat{α}}}_{i, m - 1}^{*}$ $Mathematical equation: $ G_{i,m}(\bar{\pi}_{i,m}) = g_{i,m}-\dot{\varpi}_{i,m}\hat{\alpha}^{*}_{i,m-1} - \varpi_{i,m}\dot{\hat{\alpha}}^{*}_{i,m-1} $$ , ${\bar{π}}_{i, m} = {[{\bar{\hat{ζ}}}_{i, m}, y_{r}, {\bar{y}}_{r}^{(m)}]}^{T}$ $Mathematical equation: $ \bar{\pi}_{i,m}=[\bar{\hat{\zeta}}_{i,m}, y_{r},\bar{y}^{(m)}_{r}]^{T} $$ , and ${\bar{y}}_{r}^{(m)} = {[{\dot{y}}_{r}, \dots, y_{r}^{(m)}]}^{T}$ $Mathematical equation: $ \bar{y}^{(m)}_{r}=[\dot{y}_{r}, \ldots, y^{(m)}_{r}]^{T} $$ .

Due to the fact that $G_{i, m} ({\bar{π}}_{i, m})$ $Mathematical equation: $ G_{i,m}(\bar{\pi}_{i,m}) $$ includes unknown functions, the RBF NNs $ξ_{i, m}^{* T} {\bar{R}}_{i, m} ({\bar{π}}_{i, m})$ $Mathematical equation: $ \xi^{\ast T}_{i,m}\bar{R}_{i,m}(\bar{\pi}_{i,m}) $$ are harnessed for estimate $G_{i, m} ({\bar{π}}_{i, m})$ $Mathematical equation: $ G_{i,m}(\bar{\pi}_{i,m}) $$ . As a result, we obtain

$\begin{matrix} s_{i, m} G_{i, m} ({\bar{π}}_{i, m}) \leq \frac{ϑ_{i, m} {\hat{s}}_{i, m}^{2} {\bar{R}}_{i, m}^{T} {\bar{R}}_{i, m}}{\sqrt{{\hat{s}}_{i, m}^{2} {\bar{R}}_{i, m}^{T} {\bar{R}}_{i, m} + ν_{i, m}^{2}}} + θ_{i, m} ν_{i, m} + s_{i, m} ϵ_{i, m}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} s_{i,m}G_{i,m}(\bar{\pi }_{i,m})\le \frac{\vartheta _{i,m}\hat{s}^{2}_{i,m}\bar{R}^{T}_{i,m} \bar{R}_{i,m}}{\sqrt{\hat{s}^{2}_{i,m}\bar{R}^{T}_{i,m} \bar{R}_{i,m}+\nu ^{2}_{i,m}}}+\theta _{i,m}\nu _{i,m}+s_{i,m}\epsilon _{i,m}, \end{aligned} $$$ (50)

where $ν_{i, m} = δ_{i, m} exp (- {\overset{˘}{δ}}_{i, m} t)$ $Mathematical equation: $ \nu_{i,m}=\delta_{i,m}\exp(-\breve{\delta}_{i,m}t) $$ , δ_i, m > 0 and ${\overset{˘}{δ}}_{i, m} > 0$ $Mathematical equation: $ \breve{\delta}_{i,m} > 0 $$ and $| ϵ_{i, m} ({\bar{π}}_{i, m}) | \leq ε_{i, m}$ $Mathematical equation: $ \vert \epsilon_{i,m}(\bar{\pi}_{i,m}) \vert\leq\varepsilon_{i,m} $$ .

Inserting (50) into (49), we see that

$\begin{matrix} {\dot{V}}_{i, m} (t) \leq & - s_{i, m} ϖ_{i, m} α_{i, m} + s_{i, m} ϖ_{i, m} α_{i, m} + ϑ_{i, m} ν_{i, m} + \frac{ϑ_{i, m} {\hat{s}}_{i, m}^{2} {\bar{R}}_{i, m}^{T} R_{i, m}}{\sqrt{{\hat{s}}_{i, m}^{2} {\bar{R}}_{i, m}^{T} {\bar{R}}_{i, m} + ν_{i, m}^{2}}} + {\tilde{ϑ}}_{i, m} {\dot{\hat{ϑ}}}_{i, m} \\ + {\tilde{θ}}_{i, m + 1} {\dot{\hat{ϑ}}}_{i, m + 1} + {\tilde{ρ}}_{i, m} {\dot{\hat{ρ}}}_{i, m} + s_{i, m} ϵ_{i, m} + {\tilde{W}}_{c i, m}^{T} (t) {\dot{\hat{W}}}_{c i, m} (t) + {\tilde{W}}_{a i, m}^{T} (t) {\dot{\hat{W}}}_{a i, m} (t) \\ + s_{i, m} α_{i, m + 1} - s_{i, m} \frac{1}{2} {\hat{W}}_{a i, m}^{T} (t) {\bar{R}}_{J i, m} + {\dot{V}}_{i, m - 1} . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{V}_{i,m}(t)\le &-s_{i,m}\varpi _{i,m}\alpha _{i,m} +s_{i,m}\varpi _{i,m}\alpha _{i,m}+\vartheta _{i,m}\nu _{i,m} +\frac{\vartheta _{i,m}\hat{s}^{2}_{i,m}\bar{R}^{T}_{i,m}R_{i,m}}{\sqrt{\hat{s}^{2}_{i,m}\bar{R}^{T}_{i,m}\bar{R}_{i,m} +\nu ^{2}_{i,m}}}+\tilde{\vartheta }_{i,m}\dot{\hat{\vartheta }}_{i,m}\nonumber \\&+\tilde{\theta }_{i,m+1}\dot{\hat{\vartheta }}_{i,m+1} +\tilde{\rho }_{i,m}\dot{\hat{\rho }}_{i,m}+s_{i,m} \epsilon _{i,m}+\tilde{W}^{T}_{ci,m}(t)\dot{\hat{W}}_{ci,m}(t) +\tilde{W}^{T}_{ai,m}(t)\dot{\hat{W}}_{ai,m}(t)\nonumber \\&+s_{i,m}\alpha _{i,m+1}-s_{i,m}\frac{1}{2}\hat{W}^{T}_{ai,m}(t) \bar{R}_{Ji,m}+\dot{V}_{i,m-1}. \end{aligned} $$$ (51)

In line with step l, we hold

$\begin{matrix} | s_{i, m} | (ε_{i, m} + | \frac{1}{2} {\hat{W}}_{a i, m}^{T} (t) {\bar{R}}_{J i, m} |) \leq \frac{ρ_{i, m} {\hat{s}}_{i, m}^{2}}{\sqrt{{\hat{s}}_{i, m}^{2} + ν_{i, m}^{2}}} + ρ_{i, m} ν_{i, m} . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \vert s_{i,m}\vert \Big (\varepsilon _{i,m}+\vert \frac{1}{2}\hat{W}^{T}_{ai,m}(t) \bar{R}_{Ji,m}\vert \Big )\le \frac{\rho _{i,m}\hat{s}^{2}_{i,m}}{\sqrt{\hat{s}^{2}_{i,m}+\nu ^{2}_{i,m}}}+\rho _{i,m}\nu _{i,m}. \end{aligned} $$$ (52)

Analogous to the above analysis, virtual controllers and adaptive laws is constructed as

$\begin{matrix} {\dot{\hat{ϑ}}}_{i, m} & = \frac{{\hat{s}}_{i, m}^{2} {\bar{R}}_{i, m}^{T} {\bar{R}}_{i, m}}{\sqrt{{\hat{s}}_{i, m}^{2} {\bar{R}}_{i, m}^{T} {\bar{R}}_{i, m} + ν_{i, m}^{2}}} - ν_{i, m} {\hat{ϑ}}_{i, m}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{\hat{\vartheta }}_{i,m}&=\frac{\hat{s}^{2}_{i,m} \bar{R}^{T}_{i,m}\bar{R}_{i,m}}{\sqrt{\hat{s}^{2}_{i,m}\bar{R}^{T}_{i,m} \bar{R}_{i,m}+\nu ^{2}_{i,m}}}-\nu _{i,m}\hat{\vartheta }_{i,m}, \end{aligned} $$$ (53)

$\begin{matrix} {\dot{\hat{ρ}}}_{i, m} & = \frac{{\hat{s}}_{i, m}^{2}}{\sqrt{{\hat{s}}_{i, m}^{2} + ν_{i, m}^{2}}} - ν_{i, m} {\hat{ρ}}_{i, m}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{\hat{\rho }}_{i,m}&=\frac{\hat{s}^{2}_{i,m}}{\sqrt{\hat{s}^{2}_{i,m}+\nu ^{2}_{i,m}}} - \nu _{i,m}\hat{\rho }_{i,m},\end{aligned} $$$ (54)

$\begin{matrix} α_{i, m} & = - a_{i, m} {\hat{s}}_{i, m} + \frac{N (τ_{i, m}) {\hat{θ}}_{i, m} {\hat{s}}_{i, m} {\bar{R}}_{i, m}^{T} {\bar{R}}_{i, m}}{\sqrt{{\hat{s}}_{i, m}^{2} {\bar{R}}_{i, m}^{T} {\bar{R}}_{i, m} + ν_{i, m}^{2}}} + \frac{N ({\overset{˘}{τ}}_{i, m}) {\hat{ρ}}_{i, m} {\hat{s}}_{i, m}}{\sqrt{{\hat{s}}_{i, m}^{2} + ν_{i, m}^{2}}}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \alpha _{i,m}&=-a_{i,m}\hat{s}_{i,m}+\frac{N(\tau _{i,m}) \hat{\theta }_{i,m}\hat{s}_{i,m}\bar{R}^{T}_{i,m}\bar{R}_{i,m}}{\sqrt{\hat{s}^{2}_{i,m}\bar{R}^{T}_{i,m}\bar{R}_{i,m}+\nu ^{2}_{i,m}}} +\frac{N(\breve{\tau }_{i,m})\hat{\rho }_{i,m}\hat{s}_{i,m}}{\sqrt{\hat{s}^{2}_{i,m}+\nu ^{2}_{i,m}}}, \end{aligned} $$$ (55)

where a_i, m is a positive parameter.

The formula for −s_i, mϖ_i, mα_i, m yields that

$\begin{matrix} - s_{i, m} ϖ_{i, m} α_{i, m} \leq \frac{ϑ_{i, m + 1} α_{i, m}^{2} {\hat{s}}_{i, m}^{2}}{\sqrt{{\hat{s}}_{i, m}^{2} α_{i, m}^{2} + ν_{i, m + 1}^{2}}} + ν_{i, m + 1} ϑ_{i, m + 1}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} -s_{i,m}\varpi _{i,m}\alpha _{i,m}\le \frac{\vartheta _{i,m+1}\alpha ^{2}_{i,m}\hat{s}^{2}_{i,m}}{\sqrt{\hat{s}^{2}_{i,m} \alpha ^{2}_{i,m}+\nu ^{2}_{i,m+1}}}+\nu _{i,m+1}\vartheta _{i,m+1}, \end{aligned} $$$ (56)

where $ν_{i, m + 1} = δ_{i, m + 1} exp (- {\overset{˘}{δ}}_{i, m + 1} t)$ $Mathematical equation: $ \nu_{i,m+1} = \delta_{i,m+1}\exp(-\breve{\delta}_{i,m+1}t) $$ , δ_{i, m + 1} and ${\overset{˘}{δ}}_{i, m + 1}$ $Mathematical equation: $ \breve{\delta}_{i,m+1} $$ are positive constants.

Inserting (52)–(56) into (51), it boils down to

$\begin{matrix} {\dot{V}}_{i, m} (t) \leq & s_{i, m} α_{i, m + 1} - a_{i, m} s_{i, m}^{2} + \frac{ϑ_{i, m + 1} α_{i, m}^{2} {\hat{s}}_{i, m}^{2}}{\sqrt{{\hat{s}}_{i, m}^{2} α_{i, m}^{2} + ν_{i, m + 1}^{2}}} + (N (τ_{i, m}) ϖ_{i, m}^{2} + 1) {\dot{τ}}_{i, m} - ν_{i, m} {\tilde{ϑ}}_{i, m} {\hat{ϑ}}_{i, m} \\ + (N ({\overset{˘}{τ}}_{i, m}) ϖ_{i, m}^{2} + 1) {\dot{\overset{˘}{τ}}}_{i, m} - ν_{i, m} {\tilde{ρ}}_{i, m} {\hat{ρ}}_{i, m} + {\tilde{W}}_{c i, m}^{T} (t) {\dot{\hat{W}}}_{c i, m} (t) + {\tilde{W}}_{a i, m}^{T} (t) {\dot{\hat{W}}}_{a i, m} (t) \\ + {\tilde{ϑ}}_{i, m + 1} {\dot{\hat{ϑ}}}_{i, m + 1} + ϑ_{i, m} ν_{i, m} + ν_{i, m + 1} ϑ_{i, m + 1} + {\dot{V}}_{i, m - 1} + ρ_{i, m} ν_{i, m}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{V}_{i,m}(t)\le &s_{i,m}\alpha _{i,m+1}-a_{i,m}s^{2}_{i,m} +\frac{\vartheta _{i,m+1}\alpha ^{2}_{i,m}\hat{s}^{2}_{i,m}}{\sqrt{\hat{s}^{2}_{i,m}\alpha ^{2}_{i,m}+\nu ^{2}_{i,m+1}}} +\Big (N(\tau _{i,m})\varpi ^{2}_{i,m}+1\Big )\dot{\tau }_{i,m} -\nu _{i,m}\tilde{\vartheta }_{i,m}\hat{\vartheta }_{i,m}\nonumber \\&+\Big (N(\breve{\tau }_{i,m})\varpi ^{2}_{i,m}+1\Big ) \dot{\breve{\tau }}_{i,m}-\nu _{i,m}\tilde{\rho }_{i,m} \hat{\rho }_{i,m}+\tilde{W}^{T}_{ci,m}(t)\dot{\hat{W}}_{ci,m}(t) +\tilde{W}^{T}_{ai,m}(t)\dot{\hat{W}}_{ai,m}(t)\nonumber \\&+\tilde{\vartheta }_{i,m+1}\dot{\hat{\vartheta }}_{i,m+1} +\vartheta _{i,m}\nu _{i,m}+\nu _{i,m+1}\vartheta _{i,m+1} +\dot{V}_{i,m-1}+\rho _{i,m}\nu _{i,m}, \end{aligned} $$$ (57)

where ${\dot{τ}}_{i, m} = \frac{{\hat{θ}}_{i, m} {\hat{s}}_{i, m}^{2} {\bar{R}}_{i, m}^{T} {\bar{R}}_{i, m}}{\sqrt{{\hat{s}}_{i, m}^{2} {\bar{R}}_{i, m}^{T} {\bar{R}}_{i, m} + ν_{i, m}^{2}}}$ $Mathematical equation: $ \dot{\tau}_{i,m} = \frac{\hat{\theta}_{i,m}\hat{s}^{2}_{i,m} \bar{R}^{T}_{i,m}\bar{R}_{i,m}}{\sqrt{\hat{s}^{2}_{i,m} \bar{R}^{T}_{i,m}\bar{R}_{i,m}+\nu^{2}_{i,m}}} $$ and ${\dot{\overset{˘}{τ}}}_{i, m} = \frac{{\hat{ρ}}_{i, m} {\hat{s}}_{i, m}^{2}}{\sqrt{{\hat{s}}_{i, m}^{2} + ν_{i, m}^{2}}}$ $Mathematical equation: $ \dot{\breve{\tau}}_{i,m}=\frac{\hat{\rho}_{i,m} \hat{s}^{2}_{i,m}}{\sqrt{\hat{s}^{2}_{i,m}+\nu^{2}_{i,m}}} $$ .

Setting α_{i, m + 1} and ${\dot{\hat{θ}}}_{i, m + 1}$ $Mathematical equation: $ \dot{\hat{\theta}}_{i,m+1} $$ as

$\begin{matrix} α_{i, m + 1} & = \frac{N (τ_{i, m + 1}) {\hat{ϑ}}_{i, m + 1} {\hat{s}}_{i, m} α_{i, m}^{2}}{\sqrt{{\hat{s}}_{i, m}^{2} α_{i, m}^{2} + ν_{i, m + 1}^{2}}}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \alpha _{i,m+1}&=\frac{N(\tau _{i,m+1})\hat{\vartheta }_{i,m+1} \hat{s}_{i,m}\alpha ^{2}_{i,m}}{\sqrt{\hat{s}^{2}_{i,m}\alpha ^{2}_{i,m} +\nu ^{2}_{i,m+1}}}, \end{aligned} $$$ (58)

$\begin{matrix} {\dot{\hat{ϑ}}}_{i, m + 1} & = \frac{{\hat{s}}_{i, m}^{2} α_{i, m}^{2}}{\sqrt{{\hat{s}}_{i, m}^{2} α_{i, m}^{2} + ν_{i, m + 1}^{2}}} - ν_{i, m + 1} {\hat{ϑ}}_{i, m + 1} . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{\hat{\vartheta }}_{i,m+1}&=\frac{\hat{s}^{2}_{i,m}\alpha ^{2}_{i,m}}{\sqrt{\hat{s}^{2}_{i,m}\alpha ^{2}_{i,m}+\nu ^{2}_{i,m+1}}} -\nu _{i,m+1}\hat{\vartheta }_{i,m+1}. \end{aligned} $$$ (59)

According to (43) and (57)–(59), we have

$\begin{matrix} {\dot{V}}_{i, m} (t) \leq & - \sum_{k = 1}^{m - 1} (a_{i, k} - 1) s_{i, k}^{2} - (a_{i, m} - \frac{1}{2}) s_{i, m}^{2} - \sum_{k = 1}^{m} (\frac{ν_{i, k} β_{c i, k}}{2} {\tilde{W}}_{c i, k}^{T} (t) {\bar{R}}_{J i, k} {\bar{R}}_{J i, k}^{T} {\tilde{W}}_{c i, k} (t) \\ + \frac{ν_{i, k} β_{c i, k}}{2} {\tilde{W}}_{a i, k}^{T} (t) {\bar{R}}_{J i, k} {\bar{R}}_{J i, k}^{T} {\tilde{W}}_{a i, k} (t) + \frac{ν_{i, k}}{2} {\tilde{ϑ}}_{i, k}^{2} + \frac{ν_{i, k}}{2} {\tilde{ρ}}_{i, k}^{2}) - \frac{ν_{i, m + 1}}{2} {\tilde{ϑ}}_{i, m + 1}^{2} \\ + \sum_{k = 1}^{m} μ_{i, k} + \sum_{k = 1}^{m} ((N (τ_{i, k}) ϖ_{i, k}^{2} + 1) {\dot{τ}}_{i, k} + (N ({\overset{˘}{τ}}_{i, k}) ϖ_{i, k}^{2} + 1) {\dot{\overset{˘}{τ}}}_{i, k}) \\ + (N (τ_{i, m + 1}) ϖ_{i, m} + 1) {\dot{τ}}_{i, m + 1}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{V}_{i,m}(t)\le &-\sum ^{m-1}_{k=1} \Big (a_{i,k}-1\Big )s^{2}_{i,k}-\Big (a_{i,m} -\frac{1}{2}\Big )s^{2}_{i,m}-\sum ^{m}_{k=1} \Big (\frac{\nu _{i,k}\beta _{ci,k}}{2}\tilde{W}^{T}_{ci,k}(t) \bar{R}_{Ji,k}\bar{R}^{T}_{Ji,k}\tilde{W}_{ci,k}(t) \\&+\frac{\nu _{i,k}\beta _{ci,k}}{2}\tilde{W}^{T}_{ai,k}(t) \bar{R}_{Ji,k}\bar{R}^{T}_{Ji,k}\tilde{W}_{ai,k}(t) +\frac{\nu _{i,k}}{2}\tilde{\vartheta }^{2}_{i,k} +\frac{\nu _{i,k}}{2}\tilde{\rho }^{2}_{i,k}\Big )-\frac{\nu _{i,m+1}}{2} \tilde{\vartheta }^{2}_{i,m+1} \\&+\sum ^{m}_{k=1}\mu _{i,k}+\sum ^{m}_{k=1} \Big (\big (N(\tau _{i,k})\varpi ^{2}_{i,k}+1\big ) \dot{\tau }_{i,k}+\big (N(\breve{\tau }_{i,k})\varpi ^{2}_{i,k}+1\big ) \dot{\breve{\tau }}_{i,k}\Big ) \\&+\Big (N(\tau _{i,m+1})\varpi _{i,m}+1\Big )\dot{\tau }_{i,m+1}, \end{aligned} $$$

where ${\dot{τ}}_{i, m + 1} = \frac{{\hat{ϑ}}_{i, m + 1} {\hat{s}}_{i, m}^{2} α_{i, m}^{2}}{\sqrt{{\hat{s}}_{i, m}^{2} α_{i, m}^{2} + ν_{i, m + 1}^{2}}},$ $Mathematical equation: $ \dot{\tau}_{i,m+1}=\frac{\hat{\vartheta}_{i,m+1} \hat{s}^{2}_{i,m}\alpha^{2}_{i,m}}{\sqrt{\hat{s}^{2}_{i,m} \alpha^{2}_{i,m}+\nu^{2}_ {i,m+1}}}, $$ $μ_{i, m} = \frac{ν_{i, m}}{2} ϑ_{i, m}^{2} + \frac{ν_{i, m}}{2} ρ_{i, m}^{2} + \frac{ν_{i, m + 1}}{2} ϑ_{i, m + 1}^{2} + (\frac{ν_{i, m} β_{c i, m}}{2} + \frac{ν_{i, m} β_{a i, m}}{2}) {(W_{J i, m}^{* T} {\bar{R}}_{J i, m})}^{2} + ϑ_{i, m} ν_{i, m} + ρ_{i, m} ν_{i, m} + ϑ_{i, m + 1} ν_{i, m + 1}$ $Mathematical equation: $ \mu_{i,m}=\frac{\nu_{i,m}}{2}\vartheta^{2}_{i,m} +\frac{\nu_{i,m}}{2}\rho^{2}_{i,m}+\frac{\nu_{i,m+1}}{2}\vartheta^{2}_{i,m+1} +\big(\frac{\nu_{i,m}\beta_{ci,m}}{2}+\frac{\nu_{i,m} \beta_{ai,m}}{2}\big)(W^{*T}_{Ji,m}\bar{R}_{Ji,m})^{2} +\vartheta_{i,m}\nu_{i,m} +\rho_{i,m}\nu_{i,m}+\vartheta_{i,m+1}\nu_{i,m+1} $$ .

Theorem 4

Under Assumptions 1 and 2, we devise the controllers (12), (33), (46), (55), (58), and the updating laws (13), (21), (34), (39), (47), (53), (54), (59) for the MASs (1) to achieve that all signals are bounded, the outputs of all subsystems reach a consensus asymptotically.

Proof: Defining $V (t) = \overset{N}{\sum_{i = 1}} V_{i, m} (t)$ $Mathematical equation: $ V(t)=\overset{N}{\underset{i=1}{\sum}}V_{i,m}(t) $$ . Then, $\dot{V} (t)$ $Mathematical equation: $ \dot{V}(t) $$ is obtained by

$\begin{matrix} \dot{V} (t) \leq & \sum_{i = 1}^{N} [- \sum_{k = 1}^{m - 1} (a_{i, k} - 1) s_{i, k}^{2} - (a_{i, m} - \frac{1}{2}) s_{i, m}^{2} + \sum_{k = 1}^{m} ((N (τ_{i, k}) ϖ_{i, k}^{2} + 1) {\dot{τ}}_{i, k} \\ + (N ({\overset{˘}{τ}}_{i, k}) ϖ_{i, k}^{2} + 1) {\dot{\overset{˘}{τ}}}_{i, k}) + (N (τ_{i, m + 1}) ϖ_{i, m} + 1) {\dot{τ}}_{i, m + 1} + \sum_{k = 1}^{m} μ_{i, k}] . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{V}(t)\le &\sum ^{N}_{i=1}\Bigg [-\sum ^{m-1}_{k=1} \Big (a_{i,k}-1\Big )s^{2}_{i,k}-\Big (a_{i,m}-\frac{1}{2}\Big )s^{2}_{i,m}+\sum ^{m}_{k=1}\Big (\big (N(\tau _{i,k}) \varpi ^{2}_{i,k}+1\big )\dot{\tau }_{i,k}\\&+\big (N(\breve{\tau }_{i,k})\varpi _{i,k}^{2}+1\big ) \dot{\breve{\tau }}_{i,k}\Big )+\Big (N(\tau _{i,m+1}) \varpi _{i,m}+1\Big )\dot{\tau }_{i,m+1} +\sum ^{m}_{k=1}\mu _{i,k}\Bigg ]. \end{aligned} $$$

Let $ς = min {2 (a_{i, l} - 1), 2 (a_{i, m} - \frac{1}{2})} \geq 0$ $Mathematical equation: $ \varsigma = \min\Big\{2(a_{i,l}-1), 2\Big(a_{i,m}-\frac{1}{2}\Big)\Big\}\geq0 $$ , where l = 1, 2, …, m − 1 and i = 1, …, N.

$\begin{matrix} \dot{V} (t) \leq & \sum_{i = 1}^{N} [- \sum_{k = 1}^{m} ς s_{i, k}^{2} + \sum_{k = 1}^{m} (N (τ_{i, k}) ϖ_{i, k}^{2} + 1) {\dot{τ}}_{i, k} + \sum_{k = 1}^{m} (N ({\overset{˘}{τ}}_{i, k}) ϖ_{i, k}^{2} + 1) {\dot{\overset{˘}{τ}}}_{i, k} \\ + (N (τ_{i, m + 1}) ϖ_{i, m} + 1) {\dot{τ}}_{i, m + 1} + \sum_{k = 1}^{m} μ_{i, k}] . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{V}(t)\le &\sum ^{N}_{i=1}\Bigg [-\sum ^{m}_{k=1} \varsigma s^{2}_{i,k}+\sum ^{m}_{k=1}\big (N(\tau _{i,k}) \varpi _{i,k}^{2}+1\big )\dot{\tau }_{i,k}+\sum ^{m}_{k=1} \big (N(\breve{\tau }_{i,k})\varpi _{i,k}^{2}+1\big ) \dot{\breve{\tau }}_{i,k} \nonumber \\&+\Big (N(\tau _{i,m+1})\varpi _{i,m}+1\Big ) \dot{\tau }_{i,m+1}+\sum ^{m}_{k=1}\mu _{i,k}\Bigg ]. \end{aligned} $$$ (60)

Integrating both sides of (60), one gets

$\begin{matrix} V (t) \leq & λ + \int_{0}^{t} \sum_{i = 1}^{N} [\sum_{k = 1}^{m} (N (τ_{i, k}) ϖ_{i, k}^{2} + 1) {\dot{τ}}_{i, k} + \sum_{k = 1}^{m} (N ({\overset{˘}{τ}}_{i, k}) ϖ_{i, k}^{2} + 1) {\dot{\overset{˘}{τ}}}_{i, k} \\ + (N (τ_{i, m + 1}) ϖ_{i, m} + 1) {\dot{τ}}_{i, m + 1}] d \bar{κ}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} V(t)\le &\lambda +\int ^{t}_{0}\sum ^{N}_{i=1} \Bigg [\sum ^{m}_{k=1}\big (N(\tau _{i,k})\varpi _{i,k}^{2} +1\big )\dot{\tau }_{i,k}+\sum ^{m}_{k=1} \big (N(\breve{\tau }_{i,k})\varpi _{i,k}^{2} +1\big )\dot{\breve{\tau }}_{i,k} \nonumber \\&+\Big (N(\tau _{i,m+1})\varpi _{i,m}+1\Big ) \dot{\tau }_{i,m+1}\Bigg ]d\bar{\kappa }, \end{aligned} $$$ (61)

where $λ = V (0) + \int_{0}^{t} \sum_{i = 1}^{N} (- \sum_{k = 1}^{m} ς s_{i, k}^{2}) d \bar{κ} + \sum_{k = 1}^{m} μ_{i, k}$ $Mathematical equation: $ \lambda=V(0)+\int^{t}_{0}\sum^{N}_{i=1}\Big(-\sum^{m}_{k=1}\varsigma s^{2}_{i,k}\Big)d\bar{\kappa}+\sum^{m}_{k=1}\mu_{i,k} $$ .

By Lemma 1, it follows that the variables V(t), τ_i, k, ${\overset{˘}{τ}}_{i, k}$ $Mathematical equation: $ \breve{\tau}_{i,k} $$ , τ_{i, m + 1}, $\int_{0}^{t} (N (τ_{i, k}) ϖ_{i, k}^{2} + 1) {\dot{τ}}_{i, k} d \bar{κ}$ $Mathematical equation: $ \int^{t}_{0}\big(N(\tau_{i,k})\varpi_{i,k}^{2}+1\big) \dot{\tau}_{i,k}d\bar{\kappa} $$ , as well as the integrals $\int_{0}^{t} (N ({\overset{˘}{τ}}_{i, k}) ϖ_{i, k}^{2} + 1) {\dot{\overset{˘}{τ}}}_{i, k} d \bar{κ}$ $Mathematical equation: $ \int^{t}_{0}\big(N(\breve{\tau}_{i,k})\varpi^{2}_{i,k}+1\big) \dot{\breve{\tau}}_{i,k}d\bar{\kappa} $$ , $\int_{0}^{t} (N (τ_{i, m + 1}) ϖ_{i, m} + 1) {\dot{τ}}_{i, m + 1} d \bar{κ}$ $Mathematical equation: $ \int^{t}_{0}\big(N(\tau_{i,m+1})\varpi_{i,m}+1\big) \dot{\tau}_{i,m+1}d\bar{\kappa} $$ and $\int_{0}^{\infty} \sum_{i = 1}^{N} \sum_{k = 1}^{m} ς s_{i, k}^{2} d \bar{κ}$ $Mathematical equation: $ \int^{\infty}_{0}\sum^{N}_{i=1}\sum^{m}_{k=1}\varsigma s^{2}_{i,k}d\bar{\kappa} $$ must be bounded on [0, t_a) with k = 1, …, m. According to [34], we have t_a = +∞. Based on the above results, it can be obtained that ϑ_i, k, ${\tilde{ρ}}_{i, k}$ $Mathematical equation: $ \tilde{\rho}_{i,k} $$ , ϑ_{i, m + 1}, ${\tilde{W}}_{c i, k}$ $Mathematical equation: $ \tilde{W}_{ci,k} $$ and ${\tilde{W}}_{a i, k}$ $Mathematical equation: $ \tilde{W}_{ai,k} $$ are all uniformly bounded. Furthermore, using the definitions ${\tilde{ϑ}}_{i, k} = {\hat{ϑ}}_{i, k} - ϑ_{i, k}$ $Mathematical equation: $ \tilde{\vartheta}_{i,k}=\hat{\vartheta}_{i,k}-\vartheta_{i,k} $$ , ${\tilde{ρ}}_{i, n} = {\hat{ρ}}_{i, n} - ρ_{i, n}$ $Mathematical equation: $ \tilde{\rho}_{i,n}=\hat{\rho}_{i,n}-\rho_{i,n} $$ , ${\tilde{ϑ}}_{i, m + 1} = {\hat{ϑ}}_{i, m + 1} - ϑ_{i, m + 1}$ $Mathematical equation: $ \tilde{\vartheta}_{i,m+1}=\hat{\vartheta}_{i,m+1}-\vartheta_{i,m+1} $$ , ${\tilde{W}}_{c i, k} (t) = {\hat{W}}_{c i, k} - W_{J i, k}^{*}$ $Mathematical equation: $ \tilde{W}_{ci,k}(t)=\hat{W}_{ci,k}-W^{*}_{Ji,k} $$ and ${\tilde{W}}_{a i, k} (t) = {\hat{W}}_{a i, k} - W_{J i, k}^{*}$ $Mathematical equation: $ \tilde{W}_{ai,k}(t)=\hat{W}_{ai,k}-W^{*}_{Ji,k} $$ , it follows that ${\hat{ϑ}}_{i, k}$ $Mathematical equation: $ \hat{\vartheta}_{i,k} $$ , ${\hat{ρ}}_{i, k}$ $Mathematical equation: $ \hat{\rho}_{i,k} $$ , ${\hat{ϑ}}_{i, m + 1}$ $Mathematical equation: $ \hat{\vartheta}_{i,m+1} $$ , ${\hat{W}}_{c i, k}$ $Mathematical equation: $ \hat{W}_{ci,k} $$ and ${\hat{W}}_{a i, k}$ $Mathematical equation: $ \hat{W}_{ai,k} $$ are also bounded. The boundedness of ζ_i, k follows directly from the boundedness the synchronization error. Finally, based on (44), we obtain that the system input u_i is bounded.

According to (61), $\int_{0}^{\infty} \sum_{i = 1}^{N} \sum_{k = 1}^{m} ς s_{i, k}^{2} d \bar{κ}$ $Mathematical equation: $ \int^{\infty}_{0}\sum^{N}_{i=1}\sum^{m}_{k=1}\varsigma s^{2}_{i,k}d\bar{\kappa} $$ and $\int_{0}^{\infty} \sum_{i = 1}^{N} \sum_{k = 1}^{m} ς {\hat{s}}_{i, k}^{2} d \bar{κ}$ $Mathematical equation: $ \int^{\infty}_{0}\sum^{N}_{i=1}\sum^{m}_{k=1}\varsigma \hat{s}^{2}_{i,k}d\bar{\kappa} $$ are bounded. Then, it follows that the derivative of $\sum_{i = 1}^{N} \sum_{k = 1}^{m} ς s_{i, k}^{2}$ $Mathematical equation: $ \sum^{N}_{i=1}\sum^{m}_{k=1}\varsigma s^{2}_{i,k} $$ is bounded. Further, by employing Barbalat’s Lemma, we can derive lim_{t → ∞}s_i, 1(t)=0 with i = 1, 2, …, N. Then, we have $lim_{t \to \infty} {\hat{s}}_{i, 1} (t) = 0$ $Mathematical equation: $ \lim_{t\rightarrow\infty}\hat{s}_{i,1}(t)=0 $$ . According to (5) and [35], one has

$\begin{matrix} {\dot{\hat{z}}}_{∙, 1} = - ℘ ({\hat{z}}_{∙, 1} + {\hat{s}}_{∙, 1}), \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{\hat{z}}_{\bullet ,1} = -\wp (\hat{z}_{\bullet ,1}+\hat{s}_{\bullet ,1}), \end{aligned} $$$

where ${\hat{s}}_{∙, 1} = {({\hat{s}}_{1, 1}, {\hat{s}}_{2, 1}, \dots, {\hat{s}}_{N, 1})}^{T}$ $Mathematical equation: $ \hat{s}_{\bullet, 1} = \left(\hat{s}_{1,1}, \hat{s}_{2,1}, \ldots, \hat{s}_{N,1}\right)^{T} $$ , ${\hat{z}}_{∙, 1} = {({\hat{z}}_{1, 1}, {\hat{z}}_{2, 1}, \dots, {\hat{z}}_{N, 1})}^{T}$ $Mathematical equation: $ \hat{z}_{\bullet,1} = (\hat{z}_{1,1}, \hat{z}_{2,1}, \ldots, \hat{z}_{N,1})^{T} $$ and ${\hat{x}}_{∙, 1} = {({\hat{ζ}}_{1, 1}, {\hat{ζ}}_{2, 1}, \dots, {\hat{ζ}}_{N, 1})}^{T}$ $Mathematical equation: $ \hat{x}_{\bullet,1} = (\hat{\zeta}_{1,1}, \hat{\zeta}_{2,1}, \ldots, \hat{\zeta}_{N,1})^{T} $$ .

According to graph theory, since the communication topology includes a spanning tree, ℘ has a zero eigenvalue with the corresponding eigenvector being 1_N. The other eigenvalues of ℘ lie in the open right half-plane.

By matrix transformation, we can get

$\begin{matrix} ℘ = P γ P^{- 1}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \wp =P\gamma P^{-1}, \end{aligned} $$$

where P is a matrix that is definitively positive, $γ = diag (0, \overset{˘}{λ})$ $Mathematical equation: $ \gamma = \mathrm{diag}(0,\breve{\lambda}) $$ is the Jordan canonical form of ℘ and γ P⁻¹ = [0_1 × N, Π_{(N − 1)×N}]. By defining $η = P^{- 1} {\hat{z}}_{∙, 1}$ $Mathematical equation: $ \eta = P^{-1}\hat{z}_{\bullet,1} $$ , $\dot{η}$ $Mathematical equation: $ \dot{\eta} $$ is calculated as

$\begin{matrix} \dot{η} = - γ η - γ P^{- 1} {\hat{s}}_{∙, 1} . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{\eta } = -\gamma \eta -\gamma P^{-1}\hat{s}_{\bullet ,1}. \end{aligned} $$$ (62)

According to (62), we can get ${\dot{η}}_{1} = 0$ $Mathematical equation: $ \dot{\eta}_{1}=0 $$ and η₁(t)=η₁(0) where η₁ is the first entry of η.

By defining $\overset{˘}{η} = {[η_{2}, \dots, η_{N}]}^{T}$ $Mathematical equation: $ \breve{\eta}=[\eta_{2}, \ldots, \eta_{N}]^{T} $$ and $B = diag (\overset{˘}{λ})$ $Mathematical equation: $ B=\mathrm{diag}(\breve{\lambda}) $$ , the derivative of $\overset{˘}{ζ}$ $Mathematical equation: $ \breve{\zeta} $$ is

$\begin{matrix} \dot{\overset{˘}{ζ}} = - B \overset{˘}{ζ} - Π {\hat{s}}_{∙, 1} . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \dot{\breve{\zeta }}=-B\breve{\zeta }-\mathrm \Pi \hat{s}_{\bullet ,1}. \end{aligned} $$$

Given that B > 0 and $| | {\hat{s}}_{∙, 1} | |$ $Mathematical equation: $ \vert\vert \hat{s}_{\bullet,1}\vert\vert $$ is bounded, it follows that ||η|| is also bounded. Construct Lyapunov function as $V_{\overset{˘}{ζ}} = {\overset{˘}{η}}^{T} Q \overset{˘}{ζ}$ $Mathematical equation: $ V_{\breve{\zeta}}=\breve{\eta}^{T}Q\breve{\zeta} $$ , where Q represents the solution of B^TQ + Q B = −2I. Further, we have ${\dot{V}}_{\overset{˘}{ζ}} \leq - {\overset{˘}{ζ}}^{T} \overset{˘}{η} + 2 | | Q Π | | | | {\hat{s}}_{∙, 1} {| |}^{2}$ $Mathematical equation: $ \dot{V}_{\breve{\zeta}}\leq-\breve{\zeta}^{T}\breve{\eta}+2\vert\vert Q\mathrm{\Pi}\vert\vert \vert\vert \hat{s}_{\bullet,1}\vert\vert^{2} $$ . Then, we get

$\begin{matrix} \int_{0}^{\infty} {\overset{˘}{ζ}}^{T} \overset{˘}{η} d \bar{κ} \leq - V_{\overset{˘}{η}} (t) + V_{\overset{˘}{ζ}} (0) + 2 | | Q Π | | \int_{0}^{\infty} | | {\hat{s}}_{∙, 1} {| |}^{2} d \bar{κ} . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \int ^{\infty }_{0}\breve{\zeta }^{T}\breve{\eta }d\bar{\kappa } \le -V_{\breve{\eta }}(t)+V_{\breve{\zeta }}(0)+2\vert \vert Q\mathrm \Pi \vert \vert \int ^{\infty }_{0}\vert \vert \hat{s}_{\bullet ,1}\vert \vert ^{2}d\bar{\kappa }. \end{aligned} $$$

Since $\int_{0}^{\infty} \sum_{i = 1}^{N} \sum_{k = 1}^{m} ς {\hat{s}}_{i, k}^{2} d \bar{κ}$ $Mathematical equation: $ \int^{\infty}_{0}\sum^{N}_{i=1}\sum^{m}_{k=1}\varsigma \hat{s}^{2}_{i,k}d\bar{\kappa} $$ is bounded, we get that $\int_{0}^{t} {\overset{˘}{ζ}}^{T} \overset{˘}{ζ} d \bar{κ}$ $Mathematical equation: $ \int^{t}_{0}\breve{\zeta}^{T}\breve{\zeta}d\bar{\kappa} $$ is bounded. By using Barbalat’s Lemma, we know $lim_{t \to \infty} | | \overset{˘}{η} | | = 0$ $Mathematical equation: $ \lim_{t\rightarrow\infty}\vert\vert \breve{\eta}\vert\vert=0 $$ . Since the first column of P is 1_N, we have $lim_{t \to \infty} {\hat{η}}_{∙, 1} = 1_{N} p^{T} {\hat{η}}_{∙, 1} (0)$ $Mathematical equation: $ \lim_{t\rightarrow\infty}\hat{\eta}_{\bullet,1} = 1_{N}p^{T}\hat{\eta}_{\bullet,1}(0) $$ where p is the first row of P⁻¹. Further, we can derive the output will reach consensus asymptotically. Then, $lim_{t \to \infty} ({\hat{ζ}}_{i, 1} - {\hat{ζ}}_{j, 1}) = 0$ $Mathematical equation: $ \lim_{t\rightarrow\infty}(\hat{\zeta}_{i,1}-\hat{\zeta}_{j,1})=0 $$ and lim_{t → ∞}(ζ_i, 1 − ζ_j, 1)=0 are achieved.

Remark 5 By constructing a special synchronization error, this article can achieve asymptotic output consensus of MASs even if the attackers send false state information. The strategies for deceptive attacks are successfully broadened from addressing the stabilization challenge of a single system to tackling the consensus control issue in MASs.

Remark 6 The existing adaptive control works [29–31] are confined to address the stabilization problem of the individual system and are only capable of reaching a small neighborhood around the origin for the system state. Through further improvement, our proposed control method can achieve asymptotic output consensus for MASs and is more suited for high-precision practical applications.

Figure 1.

The communication topology

Figure 2.

The system outputs ζ_i, 1

Figure 3.

The system states ζ_i, 2

Remark Our current approach emphasizes attack compensation rather than explicit detection. A promising future direction is to integrate a two-layer defense: first, a residual-based detector monitors discrepancies between system measurements and model predictions to identify attacks [36, 37]; subsequently, the resilient control framework activates targeted countermeasures.

4. Illustrative example

A numerical example is considered in the simulation. The topology structure among the 1−4 agents are illustrated in Figure 1.

Figure 4.

Available compromised system states over time. (a) The available compromised system states ${\hat{ζ}}_{i, 1}$ $Mathematical equation: $ \hat{\zeta}_{i,1} $$ . (b) The available compromised system states ${\hat{ζ}}_{i, 2}$ $Mathematical equation: $ \hat{\zeta}_{i,2} $$

Figure 5.

The signals of control input u_i

The connectivity is algebraically illustrated by the following adjacency matrix:

$\begin{matrix} A = [\begin{matrix} 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{matrix}] . \end{matrix}$ $Mathematical equation: $$ \begin{aligned} \mathcal{A} =\left[ \begin{array}{ccccc} 0&0&1&1\\ 1&0&0&0\\ 0&0&0&1\\ 0&1&0&0\\ \end{array} \right]. \end{aligned} $$$

Figure 6.

The curves of $| {\hat{W}}_{c i, 1} |$ $Mathematical equation: $ \vert \hat{W}_{ci,1} \vert $$ and $| {\hat{W}}_{a i, 1} |$ $Mathematical equation: $ \vert \hat{W}_{ai,1} \vert $$

Figure 7.

The curves of $| {\hat{W}}_{c i, 2} |$ $Mathematical equation: $ \vert \hat{W}_{ci,2} \vert $$ and $| {\hat{W}}_{a i, 2} |$ $Mathematical equation: $ \vert \hat{W}_{ai,2} \vert $$

Each follower is modeled as

$\begin{matrix} d ζ_{i, 1} = ζ_{i, 2}, d ζ_{i, 2} = u_{i} - sin (ζ_{i, 1}) - 0.2 ζ_{i, 2}, \end{matrix}$ $Mathematical equation: $$ \begin{aligned} d\zeta _{i,1} = \zeta _{i,2}, \quad d\zeta _{i,2}=u_{i}-\sin (\zeta _{i,1})-0.2\zeta _{i,2}, \end{aligned} $$$ (63)

where i = 1, 2, 3, 4. For the condition of t ≥ 2, the attack weight is chosen as ${\bar{ν}}_{i, k} (t) = 2.5 + 0.4 cos (t)$ $Mathematical equation: $ \bar{\nu}_{i,k}(t)=2.5+0.4\cos(t) $$ .

The parameters are set as a_i, 1 = a_i, 2 = 4, β_a
i, 1 = β_a
i, 2 = 2, β_c
i, 1 = β_c
i, 2 = 1.2, ${\bar{m}}_{i} = 15$ $Mathematical equation: $ \bar{m}_{i}=15 $$ , m_i = 0.05, ν_i, 1 = ν_i, 2 = ν_i, 3 = 0.1exp(−0.1t) and h_i = 2.6exp(−0.1t). The initial conditions are ζ_1, 1(0)=0.5, ζ_2, 1(0)=1.2, ζ_3, 1(0)= − 1.5, ζ_4, 1(0)= − 0.8, ζ_1, 2(0)= − 1, ζ_2, 2(0)=0.5, ζ_3, 2(0)= − 1, ζ_4, 2(0)= − 0.5, ${\hat{θ}}_{i, 1} (0) = {\hat{θ}}_{i, 2} (0) = 0$ $Mathematical equation: $ \hat{\theta}_{i,1}(0)=\hat{\theta}_{i,2}(0)=0 $$ , ${\hat{θ}}_{i, 3} (0) = 1$ $Mathematical equation: $ \hat{\theta}_{i,3}(0)=1 $$ , ${\hat{ρ}}_{i, 1} (0) = {\hat{ρ}}_{i, 2} (0) = 1$ $Mathematical equation: $ \hat{\rho}_{i,1}(0)=\hat{\rho}_{i,2}(0)=1 $$ , ${\hat{W}}_{i, a 1} (0) = {\hat{W}}_{i, a 2} (0) = 0.1$ $Mathematical equation: $ \hat{W}_{i,a1}(0)=\hat{W}_{i,a2}(0)=0.1 $$ , ${\hat{W}}_{i, c 1} (0) = {\hat{W}}_{i, c 2} (0) = 0.1$ $Mathematical equation: $ \hat{W}_{i,c1}(0)=\hat{W}_{i,c2}(0)=0.1 $$ , $τ_{i, 1} (0) = {\overset{˘}{τ}}_{i, 1} (0) = 1$ $Mathematical equation: $ \tau_{i,1}(0)=\breve{\tau}_{i,1}(0)=1 $$ , $τ_{i, 2} (0) = {\overset{˘}{τ}}_{i, 2} (0) = 0$ $Mathematical equation: $ \tau_{i,2}(0)=\breve{\tau}_{i,2}(0)=0 $$ and τ_i, 3(0)=0.

Figure 8.

Comparison of ${\hat{ζ}}_{i, 1}$ $Mathematical equation: $ \hat{\zeta}_{i,1} $$ under different ${\bar{ν}}_{i, k} (t)$ $Mathematical equation: $ \bar{\nu}_{i,k}(t) $$ for t > 2. (a) ${\bar{ν}}_{i, k} (t) = 0.5 + 0.4 cos (t)$ $Mathematical equation: $ \bar{\nu}_{i,k}(t)=0.5+0.4\cos(t) $$ . (b) ${\bar{ν}}_{i, k} (t) = 1.5 + sin (t)$ $Mathematical equation: $ \bar{\nu}_{i,k}(t)=1.5+\sin(t) $$ . (c) ${\bar{ν}}_{i, k} (t) = 2.5 + 0.4 (sin (t) + cos (t))$ $Mathematical equation: $ \bar{\nu}_{i,k}(t)=2.5+0.4(\sin(t)+\cos(t)) $$ . (d) ${\bar{ν}}_{i, k} (t) = 4.5 + sin (t)$ $Mathematical equation: $ \bar{\nu}_{i,k}(t)=4.5+\sin(t) $$

Figure 2 displays the system output trajectories of the four agents. The results in Figure 3 illustrates the variation curves of state x_i, 2. Figures 4a and 4b present two graphical representations of the available compromised system states ${\hat{x}}_{i, 1}$ $Mathematical equation: $ \hat{x}_{i,1} $$ and ${\hat{x}}_{i, 2}$ $Mathematical equation: $ \hat{x}_{i,2} $$ . After 2s, the attacker causes the sensor to transmit incorrect status information by tampering with the transmitted information, which can arouse the degradation of control performance. It is shown that agents can quickly achieve consensus under deception attacks. The controller u_i are demonstrated in Figure 5. Then, the weight estimations of actor NNs and critic NNs are depicted in Figures 6 and 7, respectively. In Figures 6 and 7, the proposed control strategy will eventually get $| {\hat{W}}_{c i, 1} | = | {\hat{W}}_{a i, 1} |$ $Mathematical equation: $ \vert \hat{W}_{ci,1}\vert=\vert \hat{W}_{ai,1}\vert $$ and $| {\hat{W}}_{c i, 2} | = | {\hat{W}}_{a i, 2} |$ $Mathematical equation: $ \vert \hat{W}_{ci,2}\vert=\vert \hat{W}_{ai,2}\vert $$ . The norms of all the NN weight estimates eventually converge. According to (15), when $P_{i, 1} (t) = ({\hat{W}}_{a i, 1} (t) - {\hat{W}}_{c i, 1} (t))^{T} ({\hat{W}}_{a i, 1} (t) - {\hat{W}}_{c i, 1} (t)) = 0$ $Mathematical equation: $ P_{i,1}(t) = \big(\hat{W}_{ai,1}(t) - \hat{W}_{ci,1}(t)\big)^{T} \big(\hat{W}_{ai,1}(t) - \hat{W}_{ci,1}(t)\big) = 0 $$ , the HJB optimality condition is satisfied. Figure 8 illustrates the system convergence under different ${\bar{ν}}_{i, k} (t)$ $Mathematical equation: $ \bar{\nu}_{i,k}(t) $$ deception attacks. It can be observed that the proposed method is able to achieve convergence even under stronger attacks.

5. Conclusions

This article has presented the optimized adaptive consensus control strategy designed for the nonlinear MASs amid deception attacks. The optimized adaptive controller, designed by incorporating Nussbaum technique, has been developed by addressing the influence of deception attacks containing time-varying gain. Taking a special synchronization error within the control scheme, the leaderless MASs can achieve asymptotic output consensus. Finally, the simulation outcomes have validated the efficacy of the suggested protocol.

Acknowledgments

We would like to thank all editors and reviewers who helped us improve the paper.

Funding

This work is supported by in part by the National Natural Science Foundation of China (62433014, 62088101), in part by the Shanghai International Science and Technology Cooperation Project (21550760900, 22510712000), in part by Shanghai Key Laboratory of Wearable Robotics and Human-Machine Interaction and in part by the Fundamental Research Funds for the Central Universities.

Conflicts of interest

The authors declare no conflicts of interest.

Data availability statement

No data are associated with this article.

Author contribution statement

Zhixu Du wrote and structured the paper, and carried out the theoretical derivations, inspections, and simulation experiments. Hao Zhang mainly surveyed the related work and jointly wrote the paper. Zhuping Wang and Sheng Gao discussed the recent developments, corrected typos, and jointly wrote the paper.

References

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Zhixu Du received the B.Sc. degree in automation from Shanxi Datong University, Datong, China, in 2018, and the M.S. degree in control theory and control engineering from Bohai University, Jinzhou, China, in 2022. He is currently pursuing his Ph.D. degree in the School of Shanghai Research Institute for Intelligent Autonomous Systems from Tongji University, Shanghai, China. His research interests include multi-agent systems, adaptive fuzzy/neural control, and distributed predictive control.

Hao Zhang received the B.Sc. degree in automatic control from Wuhan University of Technology, Wuhan, China, in 2001 and received Ph.D. degree in control theory and control engineering from Huazhong University of Science and Technology, Wuhan, China, in 2007. Currently, she is a Professor with the School of Electronics and Information Engineering, Tongji University, Shanghai, China. From December 2011 to December 2013, she was a Postdoctoral Fellow with the City University of Hong Kong. Her research interests include network-based control systems and multi-agent systems.

Zhuping Wang received the B.Eng. degree in electrical technology and the M.Eng. degree in electrical drive and automation from the Department of Automatic Control, Northwestern Polytechnic University, Xi’an, China, in 1994 and 1997, respectively, and the Ph.D. degree in intelligent robot from the National University of Singapore, Singapore, in 2003. She is currently a Professor with the School of Electronics and Information Engineering, Tongji University, Shanghai, China. Her current research interests include intelligent control of robotic systems, self-driving vehicles, and multi-agent systems.

Sheng Gao received the B.Sc. degree in automation from Donghua University, Shanghai, China in 2019, and the Ph.D. degree in control science and engineering from Tongji University, Shanghai, China, in 2025. He is currently a Postdoctoral Fellow with the School of Information Science and Engineering, East China University of Science and Technology, Shanghai, China. From January to March 2024, he was a Visiting Ph.D. Student with the Chair of Intelligent Control Systems, RWTH Aachen University, Aachen, Germany. His current research interests include optimal control, cyber-physical systems, robot and cyber security.

All Figures

	Figure 1. The communication topology
In the text

	Figure 2. The system outputs ζ_i, 1
In the text

	Figure 3. The system states ζ_i, 2
In the text

	Figure 4. Available compromised system states over time. (a) The available compromised system states ${\hat{ζ}}_{i, 1}$ $Mathematical equation: $ \hat{\zeta}_{i,1} $$ . (b) The available compromised system states ${\hat{ζ}}_{i, 2}$ $Mathematical equation: $ \hat{\zeta}_{i,2} $$
In the text

	Figure 5. The signals of control input u_i
In the text

	Figure 6. The curves of $\| {\hat{W}}_{c i, 1} \|$ $Mathematical equation: $ \vert \hat{W}_{ci,1} \vert $$ and $\| {\hat{W}}_{a i, 1} \|$ $Mathematical equation: $ \vert \hat{W}_{ai,1} \vert $$
In the text

	Figure 7. The curves of $\| {\hat{W}}_{c i, 2} \|$ $Mathematical equation: $ \vert \hat{W}_{ci,2} \vert $$ and $\| {\hat{W}}_{a i, 2} \|$ $Mathematical equation: $ \vert \hat{W}_{ai,2} \vert $$
In the text

Figure 8.

In the text

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[R1] Zhang D, Shen YP and Zhou SQ et al. Distributed secure platoon control of connected vehicles subject to DoS attack: theory and application. IEEE Trans Syst Man Cybern Syst 2021; 51: 7269–7278. [Google Scholar]

[R2] Hong H, Yu C and Yu W. Adaptive fixed-time control for attitude consensus of disturbed multi-spacecraft systems with directed topologies. IEEE Trans Network Sci Eng 2022; 9: 1451–1461. [Google Scholar]

[R3] Du Z, Zhang H and Wang Z et al. Fuzzy model predictive formation maneuver control of Multi-UAVs with interval output-constrained. IEEE Trans Fuzzy Syst 2025; 33: 1106–1119. [Google Scholar]

[R4] Lin G, Li H and Ahn CK et al. Event-based finite-time neural control for human-in-the-loop UAV attitude systems. IEEE Trans Neural Networks Learn Syst 2023; 34: 10387–10397. [Google Scholar]

[R5] Du Z, Zhang H and Wang Z et al. Model predictive formation tracking-containment control for multi-UAVs with obstacle avoidance. IEEE Trans Syst Man Cybern Syst 2024; 54: 3404–3414. [Google Scholar]

[R6] Wu Y, Chen M and Li H et al. Event-triggered-based adaptive NN cooperative control of six-rotor UAVs with finite-time prescribed performance. IEEE Trans Autom Sci Eng 2024; 21: 1867–1877. [Google Scholar]

[R7] Yang B, Li H and Yao D et al. DO-based adaptive consensus control for multiple MUAVs with dynamic constraints. IEEE Trans Syst Man Cybern Syst 2023; 53: 2387–2398. [Google Scholar]

[R8] Wu Y, Ma H and Chen M et al. Observer-based fixed-time adaptive fuzzy bipartite containment control for multiagent systems with unknown hysteresis. IEEE Trans Fuzzy Syst 2022; 30: 1302–1312. [Google Scholar]

[R9] Chen M, Yan H and Zhang H et al. Event-triggered consensus of multiagent systems with time-varying communication delay. IEEE Trans Syst Man Cybern Syst 2022; 52: 2706–2720. [Google Scholar]

[R10] Duan S, Chen G and Ren H et al. Data-driven bipartite leader-following consensus control for nonlinear multi-agent systems under hybrid attacks. Int J Robust Nonlinear Control 2024; 34: 3318–3334. [Google Scholar]

[R11] Bazzi A and Chafii M. Secure full duplex integrated sensing and communications. IEEE Trans Inf Forensics Security 2023; 19: 2082–2097. [Google Scholar]

[R12] Shen Q, Shi P and Zhu J et al. Neural networks-based distributed adaptive control of nonlinear multiagent systems. IEEE Trans Neural Networks Learn Syst 2019; 31: 1010–1021. [Google Scholar]

[R13] Liang H, Liu G and Zhang H et al. Neural-network-based event-triggered adaptive control of nonaffine nonlinear multiagent systems with dynamic uncertainties. IEEE Trans Neural Networks Learn Syst 2021; 32: 2239–2250. [Google Scholar]

[R14] Pan Y, Li Q and Liang H et al. A novel mixed control approach for fuzzy systems via membership functions online learning policy. IEEE Trans Fuzzy Syst 2022; 30: 3812–3822. [CrossRef] [Google Scholar]

[R15] Zhao W, Liu H and Wan Y et al. Data-driven formation control for multiple heterogeneous vehicles in air-ground coordination. IEEE Trans Control Network Syst 2022; 9: 1851–1862. [Google Scholar]

[R16] Wu B, Chen K and Wang D et al. Spacecraft attitude takeover control by multiple microsatellites using differential game. IEEE Trans Control Network Syst 2024; 11: 474–485. [Google Scholar]

[R17] Bhasin S, Kamalapurkar R and Johnson M et al. A novel actor-critic-identifier architecture for approximate optimal control of uncertain nonlinear systems. Automatica 2013; 49: 82–92. [Google Scholar]

[R18] Wen G, Chen CLP and Ge SS. Simplified optimized backstepping control for a class of nonlinear strict-feedback systems with unknown dynamic functions. IEEE Trans Cybern 2021; 51: 4567–4580. [Google Scholar]

[R19] Li H, Wu Y and Chen M. Adaptive fault-tolerant tracking control for discrete-time multiagent systems via reinforcement learning algorithm. IEEE Trans Cybern 2021; 51: 1163–1174. [Google Scholar]

[R20] Liu D, Wang D and Wang FY et al. Neural-network-based online HJB solution for optimal robust guaranteed cost control of continuous-time uncertain nonlinear systems. IEEE Trans Cybern 2014; 44: 2834–2847. [Google Scholar]

[R21] Pan Y, Wu Y and Lam HK. Security-based fuzzy control for nonlinear networked control systems with DoS attacks via a resilient event-triggered scheme. IEEE Trans Fuzzy Syst 2022; 30: 4359–4368. [CrossRef] [Google Scholar]

[R22] Sun Y, Tian Z and Li M et al. Automated attack and defense framework toward 5G security. IEEE Network 2020; 34: 247–253. [Google Scholar]

[R23] Wang Z, Zhou Y and Liu H et al. ThreatInsight: innovating early threat detection through threat-intelligence-driven analysis and attribution. IEEE Trans Knowl Data Eng 2024; 36: 9388–9402. [Google Scholar]

[R24] Chai Y, Du L and Qiu J et al. Dynamic prototype network based on sample adaptation for few-shot malware detection. IEEE Trans Knowl Data Eng 2022; 35: 4754–4766. [Google Scholar]

[R25] Chai Y, Chen X and Qiu J et al. MalFSCIL: a few-shot class-incremental learning approach for malware detection. IEEE Trans Inf Forensics Security 2025; 20: 2999–3014. [Google Scholar]

[R26] Li L, Yang H and Xia Y et al. Attack detection and distributed filtering for state-saturated systems under deception attack. IEEE Trans Control Network Syst 2021; 8: 1918–1929. [Google Scholar]

[R27] Ye D and Zhang TY. Summation detector for false data-injection attack in cyber-physical systems. IEEE Trans Cybern 2019; 50: 2338–2345. [Google Scholar]

[R28] Liu S, Wei G and Song Y et al. Extended Kalman filtering for stochastic nonlinear systems with randomly occurring cyber attacks. Neurocomputing 2016; 207: 708–716. [Google Scholar]

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