Control Design Of A PDE-ODE Cascade System

In recent years,the stabilization for partial differential equation and ordinary differential equation (PDE-ODE) cascade system is a problem that deserves research in the boundary control field. Due to its theories and methods,which often infiltrate in other subject areas,it has been given widespread concern in the engineering circles.This paper mainly studies the control problem for partial differential equation and ordinary differential equation (PDE-ODE) cascade system. The basic idea is the introduction of a Voltegrra transformation and the establishing the link between the original system and target system. In the control process, we firstly construct a suitable Lyapounov energy function and prove the stabilition of the target system,then Seek a reversible Volterra transform.On this basis,we employ design mind of backstepping. In this design process,two kernels have arisen. We can prove that the kernel functions have unique solution with the method of successive approxima-tion.Then we get the sate feedback controller of the closed system and transform the system into target system. This article contents is as follows:Firstly, This paper describes boundary control situation of the distributed pa-rameter system and the main contents of this paper.Secondly, the preliminary knowledge which is necessary in the paper is given. The basic concepts of stability of PDE, Lipschitz law and some important inequation are introduced.Finally,our main task is control design of partial differential equation and ordi-nary differential equation (PDE-ODE) cascade system.As well as the robustness of the feedback law to the diffusion coefficient of the heat PDE is addressed.