Control-theoretic properties of structural acoustic models with thermal effects,I. Singular estimates

We consider a class of structural acoustics models with thermoelastic flexible wall. More precisely, the PDE system consists of a wave equation (within an acoustic chamber) which is coupled to a system of thermoelastic plate equations with rotational inertia; the coupling is strong as it is accomplished via boundary terms. Moreover, the system is subject to boundary thermal control. We show that—under three different sets of coupled (mechanical/thermal) boundary conditions—the overall coupled system inherits some specific regularity properties of its thermoelastic component, as it satisfies the same singular estimates recently established for the thermoelastic system alone. These regularity estimates are of central importance for (i) well-posedness of Differential and Algebraic Riccati equations arising in the associated optimal control problems, and (ii) existence of solutions to the semilinear initial/boundary value problem under nonlinear boundary conditions. The proof given uses as a critical ingredient a sharp trace theorem pertaining to second-order hyperbolic equations with Neumann boundary data.