Boundary stabilization of partial differential equations: Theory, algorithms, and applications

We study the boundary feedback stabilization of distributed parameter systems. Applications such as those in “smart material technology” are addressed. In particular, the following two types of problems are considered.; (1) Finite dimensional compensator design for partially observed analytic systems. Finite dimensional feedback control based on a finite element approximation and using only the information from partial observation is constructed. It is shown that this feedback control provides uniform stability (in time) for the originally unstable system. Theories are illustrated by the following examples: heat equation with boundary control and boundary observation, structurally damped plate with boundary control and point observation, and structurally damped plate with point control and point observation.; (2) Nonlinear boundary stabilization of plate equations. These equations include the Euler-Bernoulli equation, the Kirchhoff equation, and the von Kármán equation. The free dynamics of these equations is either conservative (energy preserving) or strongly stable. The goal is then to introduce dissipation on the boundary (as bending moments only) as to force explicit energy decay rates of the resulting solutions to the closed-loop systems. To prove such a stability result, some improved trace regularity was obtained for these equations. Also, the robustness of the stability with respect to the thickness of the plate was studied.