Boundary Stabilization of Parabolic Equations

This monograph presents a technique, developed by the author, to design asymptotically exponentially stabilizing finite-dimensional boundary proportional-type feedback controllers for nonlinear parabolic-type equations. The potential control applications of this technique are wide ranging in many research areas, such as Newtonian fluid flows modeled by the Navier-Stokes equations; electrically conducted fluid flows; phase separation modeled by the Cahn-Hilliard equations; and deterministic or stochastic semi-linear heat equations arising in biology, chemistry, and population dynamics modeling. The text provides answers to the following problems, which are of great practical importance:

• Designing the feedback law using a minimal set of eigenfunctions of the linear operator obtained from the linearized equation around the target state
• Designing observers for the considered control systems
• Constructing time-discrete controllers requiring only partial knowledge of the state After reviewing standard notations and results in functional analysis, linear algebra, probability theory and PDEs, the author describes his novel stabilization algorithm. He then demonstrates how this abstract model can be applied to stabilization problems involving magnetohydrodynamic equations, stochastic PDEs, nonsteady-states, and more. Boundary Stabilization of Parabolic Equations will be of particular interest to researchers in control theory and engineers whose work involves systems control. Familiarity with linear algebra, operator theory, functional analysis, partial differential equations, and stochastic partial differential equations is required.

Contenu
Preliminaries.- Stabilization of Abstract Parabolic Equations.- Stabilization of Periodic Flows in a Channel.- Stabilization of the Magnetohydrodynamics Equations in a Channel.- Stabilization of the Cahn-Hilliard System.- Stabilization of Equations with Delays.- Stabilization of Stochastic Equations.- Stabilization of Nonsteady States.- Internal Stabilization of Abstract Parabolic Systems.