Formation deployment control of multi-agent systems modeled with PDE

In this paper, the formation control problem of PDE-based multi-agent systems (MASs) is discussed. Firstly, the MASs are developed on a one-dimensional chain topology based on the polar coordinate system, and the dynamics of MASs is simulated using the spatial-varying coefficient wave equation. Secondly, a boundary control scheme is proposed by combining PDE-backstepping technique and the Volterra integral transformation. The well-posedness of kernel function is proved by using the iterative and inductive methods. Then, the stability of the closed-loop system is proved by using Lyapunov direct method. Finally, the PDE model is discretized using the finite difference method, and the distributed cooperative control protocol is obtained, in which the followers only need to know the location information of themselves and their neighbors. With this control protocol, leaders drive the MAS to stabilize in the desired formation. Both theoretical analysis and numerical simulation prove that the proposed control scheme is effective.