Boundary Control for a Kind of Coupled PDE-ODE System

A coupled system of an ordinary differential equation (ODE) and a heat partial differential equation (PDE) with spatially varying coefficients is discussed. By using the PDE backstepping method, the state-feedback stabilizing controller is explicitly constructed with the assumptions and , respectively. The closed-loop system is proved to be exponentially stable by this controller. A simulation example is presented to illustrate the effectiveness of the proposed method. 1. Introduction Predictor-feedback control design [1, 2] has been an active area of research for PDE or PDE-ODE coupled control systems [3–5] with actuator and sensor delays that have rich physical backgrounds such as coupled electromagnetic, coupled mechanical, and coupled chemical reactions. The input delays to the ODE system can be modeled with the first-order hyperbolic PDE (transport PDE) and the boundary condition . Thus, the original ODE system with input delay can be represented as the following ODE-PDE coupled system (1) that is driven by the input from the boundary of the PDE: Control design of coupled PDE-ODE systems was considered in [6–10]. The controller design based on the backstepping method for the coupled system (1) was designed in [9, 10]. More recently in [11], a heat diffusion PDE-ODE coupled system was considered, and a wave PDE-ODE coupled system was considered in [12]. The control system with interaction for this system coupled between the ODE and the PDE was considered in [13]: In this system, the ODE acts back on the PDE by the state of the ODE; meanwhile, the PDE acts on the ODE, which models the solid-gas interaction of heat diffusion and chemical reaction. In this paper, we replace the spatially constant coefficient of the PDE subsystem in (2) by the spatially varying coefficient ; that is, , which implies that the effects from the ODE subsystem to the PDE subsystem are varying with the location . In fact, control of the coupled systems is an important subject in control theory since this type of system arises frequently in control engineering. The objective of this paper is to convert a PDE-ODE coupled system into a closed-loop target system that is exponentially stable in the sense of the norm , with a designed stable state-feedback controller by using the backstepping-based predictor design method. Under the assumptions and , respectively, we further obtain the explicit expressions of the kernel function of the backstepping transformation. This paper is organized as follows. In Section 2, we propose the interaction of PDE-ODE coupled control system. In