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Generation of analytic semi-groups in L² for a class of second order degenerate elliptic operators

Cannarsa P., Rocchetti D., Vancostenoble J.

Control and Cybernetics

2008

Vol. 37, no 4

831-878

We study the generation of analytic semigroups in the L² topology by second order elliptic operators in divergence form, that may degenerate at the boundary of the space domain. Our results, that hold in two space dimensions, guarantee that the solutions of the corresponding evolution problems support integration by parts. So, this paper provides the basis for deriving Carleman type estimates for degenerate parabolic operators.

Strong decay for one-dimensional wave equations with nonmonotone boundary damping

Pierre M., Vancostenoble J.

Control and Cybernetics

2000

Vol. 29, no 2

473-484

This paper is a contribution to the following question : consider the classical wave equation damped by a nonlinear feedback control which is only assumed to decrease the energy. Then, do solutions to the perturbed system still exist for all time? Does strong stability occur in the sense that the energy tends to zero as time tends to infinity? We prove here that the answer to both questions is positive in the specific case of the one-dimensional wave equation damped by boundary controls which are functions of the observed velocity. The main point is that no monotonicity assumption is made on the damping term.