In this article, the consensus, collision avoidance, and distributed convex optimization problems of a class of multi-agent systems with a time-varying objective function are studied. Moreover, the proposed method in this article can reduce the chattering problem of the controller input caused by the sign function. First, the sign function is used to design the controller to improve the convergence rate of the system, on this basis, the optimization item is designed for the more complex time-varying objective function to ensure that the value of the objective function is the smallest when the system achieves the speed consensus. Second, considering the chattering problem caused by the sign function, the designed controller is improved to reduce the chattering phenomenon.Third, two novel Lyapunov functions are constructed to verify the stability of the system and the distributed optimization objective. Due to the controller is designed for the time-varying objective function, moreover, the tasks of the formation and collision avoidance are needed to take into account, therefore, the designed Lyapunov functions in this article are more difficult. Finally, the formation function is designed to ensure that the multi-agent system achieves the desired formation configuration. The simulation results show the correctness of the theoretical design of this article and the superiority in extending the running time of the system.
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