###### Abstract  
This paper addresses the distributed optimal decoupling synchronous control of multiple autonomous unmanned linear systems (MAUS) subject to complex network dynamic coupling. The leader-follower mechanism based on neighborhood error dynamics is established and the network coupling term is regarded as the external disturbance to realize the decoupling cooperative control of each agent. The Bounded \(L_{2}\)-Gain problem for the network coupling term is formulated into a multi-player zero-sum differential game. It is shown that the solution to the multi-player zero-sum differential game requires the solution to coupled Hamilton-Jacobi (HJ) equations. The coupled HJ equations are transformed into an algebraic Riccati equation (ARE), which can be solved to obtain the Nash equilibrium of a multi-player zero-sum game. It is shown that the bounded \(L_{2}\)-Gain for coupling attenuation can be realized by applying the zero-sum game solution as the control protocol and the ultimately uniform boundedness (UUB) of a local neighborhood error vector under conservative conditions is proved. A simulation example is provided to show the effectiveness of the proposed method. synchronous control; multiple autonomous unmanned linear systems; bounded \(L_{2}\)-gain; multi-player zero-sum game; coupling attenuation; coupled Hamilton-Jacobi equation; nash equilibrium; algebraic Riccati equation +
Footnote †: journal: Applied  
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Footnote †: journal: Applied  
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Footnote 1: institutetext: Applied Mathematics and Computation, 10  
result of coupling from adjacent node agents, which brings considerable complexity to the design of the control protocol. Ref. [6] formulated this intricate relationship as a differential game, namely, a multi-agent differential graphical game, by defining a local performance index for each agent. Optimal control and game theory [7; 8] has been successful utilized to formulate strategic behavior, where the dynamic of each agent relies on the actions of itself and its neighbors. In an optimal control and differential graphical game, each agent minimizes the performance objective (cost value function) by adjusting its control strategy to optimal. In [9], the finite-time optimal coordination problem of multi-agent systems (MASs) is investigated. The authors in [10] addressed the centralized optimal coordination problem under tree formation constraints. In [11], the robust optimal formation control of heterogeneous MASs is studied. These published graphical games and optimal control methods for consensus and synchronization are achieved based on the solution of coupled Hamilton-Jacobi-Isaacs (HJI) equations and Hamilton-Jacobi-Bellman (HJB) equations, respectively. In practice, coupled HJI and coupled HJB equation are difficult to be solved by analytical methods due to their inherent nonlinearity and uncertainty. The reinforcement learning (RL) method is often regarded as the effective method to solve the coupled HJI and coupled HJB equation. RL is the branch of machine learning concerned with how to methodically adjust the control strategy of agents based on the rewards from the environment [12; 13; 14; 15]. In [16], an online distributed optimal adaptive algorithm is proposed for a differential graphical game, the intelligent identifier is designed to find the unknown dynamic and the neural actor-critic network structure is introduced to find the solutions of the Nash equilibrium. In [17], the bounded \(L_{2}\)-gain consensus problem for the MASs with external disturbance is formulated into the zero-sum differential game by introducing a specific performance index and a policy iteration (PI) algorithm-based RL is provided to find the solution to the coupled HJI equations. In [18], the optimal synchronization control problem is studied for homogeneous MASs with input saturation by using the RL methods.