No abstract

Abstract. We show herein the uniform stability of a thermoelastic plate model with no added dissipative mechanism on the boundary (uniform stability of a thermoelastic plate with added boundary dissipation was shown in [J. LAGNESE, Boundary Stabilization of Twin Plates, SIAM Stud. Appl. Math. 10, SIAM, Philadelphia, PA, 1989], as was that of the analytic case-where rotational forces are neglected-in [Z. LIU and S. ZHENG, Quarterly Appl. Math., 55 (1997), pp. 551-564]). The proof is constructive in the sense that we make use of a multiplier with respect to the coupled system involved so as to generate a fortiori the desired estimates; this multiplier is of an operator theoretic nature, as opposed to the more standard differential quantities used for related work. Moreover, the particular choice of our multiplier becomes clear only after recasting the PDE model into an associated abstract evolution equation.

In this paper, we provide results concerning the optimal feedback control of a system of partial dierential equations which arises within the context of modelling a particular uid/structure interaction system seen in structural acoustics, this application being the primary motivation for our work. This model onsists of two coupled PDE's exhibiting parabolic and hyperbolic characteristics respectively; the control action, in this case, is modelled by a highly unbounded operator. We rigorously justify a optimal control theory or this class of problems and characterize the optimal control through a suitable Riccati Equation. This is achieved, in part, by exploiting recent techniques in the area of optimization of analytic systems with unbounded inputs, along with a local microanalysis of the hyperbolic part of the dynamics, an analysis which will consider the propagation of singularities and optimal \trace" behavior of the solutions.

Abstract. We show here the uniform stabilization of a coupled system of hyperbolic and parabolic PDE's which describes a particular fluid/structure interaction system. This system has the wave equation, which is satisfied on the interior of a bounded domain Ω, coupled to a "parabolic-like" beam equation holding on ∂Ω, and wherein the coupling is accomplished through velocity terms on the boundary. Our result is an analog of a recent result by Lasiecka and Triggiani which shows the exponential stability of the wave equation via Neumann feedback control, and like that work, depends upon a trace regularity estimate for solutions of hyperbolic equations.

We address semigroup well-posedness of the fluid-structure interaction of a linearized compressible, viscous fluid and an elastic plate (in the absence of rotational inertia). Unlike existing work in the literature, we linearize the compressible Navier-Stokes equations about an arbitrary state (assuming the fluid is barotropic), and so the fluid PDE component of the interaction will generally include a nontrivial ambient flow profile U. The appearance of this term introduces new challenges at the level of the stationary problem. In addition, the boundary of the fluid domain is unavoidably Lipschitz, and so the well-posedness argument takes into account the technical issues associated with obtaining necessary boundary trace and elliptic regularity estimates. Much of the previous work on flow-plate models was done via Galerkin-type constructions after obtaining good a priori estimates on solutions (specifically [18]-the work most pertinent to ours here); in contrast, we adopt here a Lumer-Phillips approach, with a view of associating solutions of the fluid-structure dynamics with a C0-semigroup e At t≥0 on the natural finite energy space of initial data. So, given this approach, the major challenge in our work becomes establishing of the maximality of the operator A which models the fluid-structure dynamics. In sum: our main result is semigroup well-posedness for the fully coupled fluid-structure dynamics, under the assumption that the ambient flow field U ∈ H 3 (O) has zero normal component trace on the boundary (a standard assumption with respect to the literature). In the final sections we address well-posedness of the system in the presence of the von Karman plate nonlinearity, as well as the stationary problem associated with the dynamics.