Internal stabilization of three interconnected semilinear reaction-diffusion PDEs with one actuated state

Abstract

This work deals with the exponential stabilization of a system of three semilinear parabolic partial differential equations (PDEs), written in a strict feedforward form. The diffusion coefficients are considered distinct and the PDEs are interconnected via both a reaction matrix and a nonlinearity. Only one of the PDEs is assumed to be controlled internally, thereby leading to an underactuated system. Constructive and efficient control of such underactuated systems is a nontrivial open problem, which has been solved recently for the linear case. In this work, these results are extended to the semilinear case, which is highly challenging due the coupling introduced by the semilinearity. Modal decomposition is employed, where due to the semilinearity, the finite-dimensional part of the solution is coupled with the infinite-dimensional tail. A transformation is then employed to map the finite-dimensional part into a target system, which allows for an efficient design of a static linear proportional state-feedback controller. Furthermore, a high-gain approach is employed in order to compensate for the semilinear terms. Lyapunov stability analysis is performed, leading to LMI conditions guaranteeing exponential stability with an arbitrary decay rate. The LMIs are shown to always be feasible, provided the number of actuators and the value of the high gain parameter are large enough. Numerical examples demonstrate the proposed approach.