Stabilization Of Coupled PDE-ODE Systems

Control design for a coupled system of a parabolic partial differential equation (PDE) and an ordinary differential equation (ODE), together with control design for a coupled PDE-ODE system with input delay, are discussed in this paper. Through control design, exponential stabilization of the closed-loop systems is achieved.The method of PDE backstepping is recurred to, which has successfully solved control design problem for cascaded PDE-ODE systems. Firstly, an invertible inte-gral transformation is looked for to convert the original system into an exponentially stable target system. Secondly, a controller can be obtained through the transfor-mation. Then, with the invertibility of the transformation, exponential stability of the closed-loop system is achieved. What's more, before designing controller for the delayed coupled PDE-ODE system, the method of establishing an interconnection to transfer the input time delay into a space extension is employed to transform the system into a cascade of a PDE with a coupled PDE-ODE system without time delay. Some algebraic and analytical skills are also employed.Compared with the achievements that have been acquired, the most important feature of this paper lies in the more complicated and more practical coupled systems considered in this paper. The established results mostly discuss cascaded PDE-ODE systems, where the interactions between the PDEs and the ODEs are one-directional, that is, only the PDEs act on the ODEs. Whereas in the systems considered in this paper, the interactions are two-directional, that is, the ODEs act back on the PDEs when the PDEs act on the ODEs. What's more, input delay has also been considered. Therefore, the results obtained in this paper not only further develop the established results, but also have more obvious value in practice.