###### Abstract
This research utilizes the neural network as an approximator and a design-specific update law so that the neural network approximates the optimal value function and optimal control strategy with certain precision. However, the strict asymptotic convergence proof of the neural network is not given in these works, and only the boundedness of approximate errors is guaranteed. In addition, the neural network approximator-based RL needs to satisfy the persistence of the excitation condition (PE) [19; 20; 21; 22; 23], which also limits the practical engineering application of these methods. The quadratic optimal control problem of a linear single system can be solved by solving an algebraic Riccati equation (ARE) [24], but the optimal control problem of MASs is far more complicated than that of a single system owing to the state coupling in the control design. At present, some optimal control methods of MASs are accompanied by a huge amount of calculations and strong assumptions. Motivated by the above discussion, this paper focuses on the optimal cooperative control of Multiple Autonomous Unmanned linear systems (MAUS) from a new perspective, i.e., the adjacent nodes' input coupling part is regarded as the external disturbance. Thus, the complex distributed multi-agent error dynamics are decoupled into centralized multi-input dynamics. Inspired by the idea of a zero-sum game in [17], this paper formulates these centralized multi-input dynamics into multiple independent multi-player zero-sum differential games. The motivation is to realize the decoupled optimal synchronous control of MASs and the main contributions of this paper are listed in the following:  
1. The coupling among the distributed multi-agents is equivalent to the disturbance from different channels, and the local neighborhood error dynamics of each agent are modeled as an independent centralized multi-player game. 2. The bounded \(L_{2}\)-gain problem for coupling attenuation is introduced and is formulated into a multi-player zero-sum game by defining a modified performance index. Different from the \(L_{2}\)-gain problem of [17], concerning disturbance rejection, the motive of the bounded \(L_{2}\)-gain for the coupling attenuation studied here is to suppress the coupling effect on the performance. It is proved that the solution of the zero-sum game requires the solution of the coupled Hamilton-Jacobi (HJ) equation. The coupled HJ equation of each agent is transformed into an independent equivalent algebraic Riccati equation, which simplifies the solution process effectively. This paper is organized as follows. Section 2 provides the mathematical background and derives the local error dynamics of each node that is coupled by its own control protocol and those of its neighbors. Section 3 proposes the problem formulation of the bounded \(L_{2}\)-gain for coupling attenuation and its equivalent multi-player zero-sum differential game. Section 4 transforms this zero-sum differential game into the solution of an algebraic Riccati equation and proves the ultimately uniform boundedness of the local neighborhood error, conservatively. The simulation results and conclusion is presented in Sections 5 and 6, respectively.