Generation of analytic semi-groups in L-2 for a class of second order degenerate elliptic operators

We study the generation of analytic semigroups in the L-2 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

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

Precise estimates for biorthogonal families under asymptotic gap conditions

We consider the typical one-dimensional strongly degenerate parabolic operator Pu = ut (xux)x with 0 < x < ` and 2 (0; 2), controlled either by a boundary control acting at x = `, or by a locally distributed control. Our main goal is to study the dependence of the so-called controllability cost needed to drive an initial condition to rest with respect to the degeneracy parameter . We prove that the control cost blows up with an explicit exponential rate, as eC=((2)2T), when ! 2 and/or T ! 0+. Our analysis builds on earlier results and methods (based on functional analysis and complex analysis techniques) developed by several authors such as Fattorini-Russel, Seidman, G uichal, Tenenbaum-Tucsnak and Lissy for the classical heat equation. In particular, we use the moment method and related constructions of suitable biorthogonal families, as well as new ne properties of the Bessel functions J of large order (obtained by ordinary dierential equations techniques)

Exact controllability of an aeroacoustic model

We study the exact controllability of a fluid-structure model. The fluctuations of velocity and pressure in the fluid are described by a potential, and the structure is a membrane located in a part $\Gamma_s$ of the boundary of the domain Ω. The potential ϕ and the transverse displacement z satisfy a coupled system of two wave equations, one in the domain $\Omega\times (0,T)$, the other one in the boundary $\Gamma_s\times (0,T)$. Taking two boundary controls, the first one in a boundary condition satisfied by the potential, and the second one in a boundary condition of the structure equation, we identify the space of controllable initial conditions when the geometrical controllability conditions are satisfied. As in the case of the so-called Helmholtz fluid-structure model [10], the difficulty in the treatement of the observability inequalities, in the definition of very weak solutions, and in the proof of controllability result, comes from the coupling terms of the system. To overcome these difficulties, we show that the variant introduced in [10] of the classical Hilbert Uniqueness Method can be adapted to the aeroacoustic model we consider

The cost of controlling strongly degenerate parabolic equations

We consider the typical one-dimensional strongly degenerate parabolic operator Pu = ut - (xalphaux)x with 0 < x < l and alpha is an element of (0, 2), controlled either by a boundary control acting at x = l, or by a locally distributed control. Our main goal is to study the dependence of the so-called controllability cost needed to drive an initial condition to rest with respect to the degeneracy parameter alpha. We prove that the control cost blows up with an explicit exponential rate, as eC/((2-alpha)2T), when alpha -> 2- and/or T -> 0+. Our analysis builds on earlier results and methods (based on functional analysis and complex analysis techniques) developed by several authors such as Fattorini-Russel, Seidman, Guichal, Tenenbaum-Tucsnak and Lissy for the classical heat equation. In particular, we use the moment method and related constructions of suitable biorthogonal families, as well as new fine properties of the Bessel functions Jnu of large order nu (obtained by ordinary differential equations techniques)