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This work deals with the exponential stabilization of a system of threesemilinear parabolic partial differential equations (PDEs), written in a strictfeedforward form. The diffusion coefficients are considered distinct and thePDEs are interconnected via both a reaction matrix and a nonlinearity. Only oneof the PDEs is assumed to be controlled internally, thereby leading to anunderactuated system. Constructive and efficient control of such underactuatedsystems is a nontrivial open problem, which has been solved recently for thelinear case. In this work, these results are extended to the semilinear case,which is highly challenging due the interconnection that is introduced by thenonlinearity. Modal decomposition is employed, where due to nonlinearity, thefinite-dimensional part of the solution is coupled with theinfinite-dimensional tail. A transformation is then performed to map thefinite-dimensional part into a target system, which allows for an efficientdesign of a static linear proportional state-feedback controller. Furthermore,a high-gain approach is employed in order to compensate for the nonlilinearterms. Lyapunov stability analysis is performed, leading to LMI conditionsguaranteeing exponential stability with arbitrary decay rate. The LMIs areshown to always be feasible, provided the number of actuators and the value ofthe high gain parameter are large enough. Numerical examples show theefficiency of the proposed approach.

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