Dirichlet Boundary Stabilization of Unstable Mixed Parameter Systems

Abstract

In this paper, we study the finite-dimensional stabilization problem of the cascade consisting of the one-dimensional transport-diffusion process and an unstable Ordinary Differential Equation (ODE) plant, where the ODE plant is connected with the transport-diffusion process through a filter. The input to the whole system is only Dirichlet boundary input to the transport diffusion process, and the outputs are the Dirichlet data at the boundary of process domain and the output from the ODE plant. In this paper, we use the latest method and show that the one-dimensional transport-diffusion process with such input and output can be formulated as a system with Aγ-bounded output operator and direct feed through term. It is shown that, under the assumption that the ODE plant is controllable and observable, the finite-dimensional model of the whole system becomes controllable and observable, when the filter mentioned above is a Residual Mode Filter (RMF). This fact enables us to construct a finite-dimensional stabilizing controller by using an RMF approach.