Control spillover in dissipative evolution equations

Abstract

We consider the problem of global stabilization of a semilinear dissipative parabolic PDE by finite-dimensional control. The infinite-dimensional system is decomposed into a reduced order finite-dimensional system and a residual system. Coupling between the systems occurs through the nonlinear function and also through the actual controller; a phenomenon known as control spillover. Rather than decompose the original PDE into Fourier modes, we decompose the derivative of a control Lyapunov function. Upper bounds on the terms representing control spillover are obtained. An LQR design is used to stabilize the system with these upper bounds. Relations between system and LQR control design parameters are given to ensure global stability and robustness with respect to control spillover.