Local exponential stabilization of a coupled ODE-Fisher’s PDE system

Abstract

In this paper, we investigate the problem of exponential stabilization of a coupled control system consisting of a linear ordinary differential equation (ODE) and a nonlinear partial differential equation (PDE). The nonlinear PDE is the one-dimensional Fisher’s equation defined on a bounded domain. The control input for the whole system acts at the right boundary of the PDE domain, whereas the left boundary injects a Dirichlet term in the ODE subsystem and the coupling is given as a Robin condition at the left boundary. Based on the infinite dimensional backstepping method for the linear part, a state feedback boundary controller is explicitly derived. Under this controller, we show that the considered coupled ODE-Fisher’s PDE system is globally well-posed in the Hilbert space Rn×L2(0,1). Moreover, using a well-adapted strict Lyapunov functional, we establish its local exponential stabilization around its zero equilibrium in the topology of Rn×L2(0,1)-norm. A simulation result that reflects the effectiveness of the proposed approach is given.