Singular estimates and uniform stability of coupled systems ofhyperbolic/parabolic PDEs

Abstract

Statement of the problem. Main results1.1. Introduction; singular estimates for coupled PDE systems. The present paper sets itself along that line of research established by [1,3]-and later streamlined and simplified in the exposition of [16]-that is focused on mathematical properties of an accepted acoustic chamber model [8,31], subject to unbounded control action on its flexible wall.We consider a generalization of an established structural acoustic model. This consists of a coupled system of two partial differential equations of different types-a hyperbolic PDE acting within the acoustic chamber and a paraboliclike PDE acting on the elastic wall of the chamber-which are strongly coupled at their common interface. Unlike prior literature, we allow the parameter α-which measures the strength of damping in the plate-like component-to run over the entire range 1/2 ≤ α ≤ 1 of analyticity (parabolicity) of its free dynamics. Prior literature considered only the limit case α = 1 ("Kelvin Voight damping"). However, the limit case α = 1/2 ("structural, square root damping") is at least equally important; indeed, even more so in applications. Entirely new phenomena arise over the case α = 1 as α decreases from 1 to 1/2, which we describe below. Indeed, as α decreases from 1 to 1/2, we seek to achieve two critical control-theoretic properties of the overall coupled system: (i) a quantitatively precise singular estimate for the operator e At B as t 0 and, simultaneously, (ii) a uniform (exponential) stability result of the free dynamics operator e At (which is not analytic, of course). Here, B is the unbounded control operator: it typically contains derivatives of Dirac measures supported on points or curves of the elastic wall of the chamber. It turns out that goals (i) and (ii) are in conflict with one another, as α decreases from 1 to 1/2. Indeed, as the damping strength α of the plate (parabolic) component decreases from 1 to 1/2, a progressively stronger

Uniform Stability of a Coupled System of Hyperbolic/Parabolic PDE’s with Internal Dissipation

“…The PDE model under consideration in this paper is a nonlinear generalization of a coupled system of hyperbolic/parabolic PDEs studied in [6]. A brief mathematical description of this system is given below; see [6] for a thorough analysis of the linear problem.…”

“…A brief mathematical description of this system is given below; see [6] for a thorough analysis of the linear problem. wall.…”

“…The present paper is focused on uniform stabilization of a coupled system of hyperbolic/parabolic PDEs, which is a nonlinear generalization of a PDE model originated in [6].…”

“…The mathematical model studied in [6] consists of a wave equation in pation term of the wave component which accounts for lack of uniform stability of the overall system, even in the presence of viscous damping on the entire domain (overdamping phenomenon). In [6J it has been shown that by introducing a comparable static damping in the boundary condition of the wave component, then the corresponding feedback system is (exponentially) uniformly stable.…”

“…As a final comment we point out that it would be very interesting to study (infinite-dimensional) Hamilton-Jacobi-Bellman (HJB) Equations associated with optimal control problems for abstract equations in Banach spaces of the form y' = Ay + Bu + F(y), when (a) the control operator B is unbounded, and (b) the operator eAt B satisfies the so called "singular estimate" (cf [1,9,12,6]). Let us recall that in the special Linear Quadratic (LQ) case, recently it has been shown that in the framework defined by (a)-(b), namely if the singular estimate holds, yet in the absence of analyticity of the underlying semigroup, then wellposedness of associated Riccati equations, along with the sought after property that the gain operator is bounded, follow as well [12].…”