STABILIZATION AND STRUCTURAL ASSIGNMENT OF DIRICHLET BOUNDARY FEEDBACK PARABOLIC EQUATIONS.

A parabolic equation defined on a bounded domain is considered, with input acting on the boundary through the Dirichlet B. C. expressed as a specified finite dimensional feedback of the solution. The free system (zero B. C. ) is assumed throughout to be unstable. Two main results are established. First, a novel proof that fully solves the corresponding boundary feedback stabilization problem is provided. Most of the paper is devoted to the second problem, structural or spectral assignment, which is a natural question relevant to the selfadjoint case. Here, under the same algebraic condition plus mild extra conditions the authors establish the existence of boundary vectors that yield a more refined and stronger result for the corresponding feedback solutions, in the form of the following desirable structural or spectral property: for positive times, the feedback solutions, in the form of the following desirable structural or spectral property: for positive times, the feedback solutions can be expressed as an infinite linear combination of decaying exponentials. A semigroup approach is employed for both problems, but the corresponding technique of solution are vastly different