  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 interconnection that is introduced by the
nonlinearity. Modal decomposition is employed, where due to nonlinearity, the
finite-dimensional part of the solution is coupled with the
infinite-dimensional tail. A transformation is then performed 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 nonlilinear
terms. Lyapunov stability analysis is performed, leading to LMI conditions
guaranteeing exponential stability with 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 show the
efficiency of the proposed approach.
