Internal controller design for semilinear parabolic systems

This paper introduces an optimal controller design methodology for semilinear parabolic systems which is based on the finite-element approximation of the linearized system. The feedback controller is finite-dimensional, with dimension given by the largest algebraic multiplicity of the unstable eigenvalues of the linearized system, has support in an arbitrary nonempty open subset and is active in one equation only. It is shown theoretically and by numerical simulations that the proposed controllers locally exponentially stabilize the steady-state solution of semilinear parabolic systems.