Low-Gain Compensation for PDE-ODE Cascade Systems with Distributed Diffusion and Counter-Convection

This note proposes two low-gain controllers for systems with distributed input dynamics, governed by diffusion equations with counter convection. It is shown that the considered PDE-ODE cascade system can be exponentially stabilized with the proposed low-gain controllers as long as the open-loop plant is not exponentially unstable. A key advantage of the proposed controllers is their much simplified form in comparison with the existing backstepping-designed controllers developed for the same systems. However at the same time, additional restrictions on the eigenvalues of the open-loop plant have to be imposed for low-gain controllers, which are not required in the backstepping design. Moreover, the Robin boundary condition, which is more general and also more difficult to deal with, is considered in this work. Several simulation examples are provided to illustrate effectiveness of our controllers.