George Avalos

Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation

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 Thin 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. Z. Heng, 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.

Higher Regularity of a Coupled Parabolic-Hyperbolic Fluid-Structure Interactive System

Abstract This paper considers an established model of a parabolic-hyperbolic coupled system of two PDEs, which arises when an elastic structure is immersed in a fluid. Coupling occurs at the interface between the two media. Semigroup well-posedness on the space of finite energy for {𝑤, 𝑤𝑡, 𝑢} was established in [Contemp. Math. 440: 15–54, 2007]. Here, [𝑤, 𝑤𝑡] are the displacement and the velocity of the structure, while 𝑢 is the velocity of the fluid. The domain D(A) of the generator A does not carry any smoothing in the 𝑤-variable (its resolvent 𝑅(λ, A) is not compact on this component space). This raises the issue of higher regularity of solutions. This paper then shows that the mechanical displacement, fluid velocity, and pressure terms do enjoy a greater regularity if, in addition to the I.C. {𝑤0, 𝑤1, 𝑢0} ∈ D(A), one also has 𝑤0 in (𝐻2(Ω𝑠))𝑑.

Boundary Controllability of Thermoelastic Plates via the Free Boundary Conditions

Controllability properties of a partial differential equation (PDE) model describing a thermoelastic plate are studied. The PDE is composed of a Kirchoff plate equation coupled to a heat equation on a bounded domain, with the coupling taking place on the interior and boundary of the domain. The coupling in this PDE is parameterized by

Exact controllability of structural acoustic interactions

\alpha >0

Uniform decay properties of a model in structural acoustics

. Boundary control is exerted through the (two) free boundary conditions of the plate equation and through the Robin boundary condition of the temperature. These controls have the physical interpretation of inserted forces and moments and prescribed temperature, respectively, all of which act on the edges of the plate. The main result here is that under such boundary control, and with initial data in the basic space of well-posedness, one can simultaneously control the displacement of the plate exactly and the temperature approximately. Moreover, the thermal control may be taken to be arbitrarily smooth in time and space, and the thermal control region may be any nonempty subset of the boundary. This controllability holds for arbitrary values of the coupling parameter

Lack of Time-Delay Robustness for Stabilization of a Structural Acoustics Model

\alpha

Uniform Stability of Nonlinear Thermoelastic Plates with Free Boundary Conditions

, with the optimal controllability time in line with that seen for uncoupled Kirchoff plates.

Boundary controllability of a coupled wave/Kirchoff system

Abstract In this article, we work to discern exact controllability properties of two coupled wave equations, one of which holds on the interior of a bounded open domain Ω , and the other on a segment Γ0 of the boundary ∂Ω . Moreover, the coupling is accomplished through terms on the boundary. Because of the particular physical application involved—the attenuation of acoustic waves within a chamber by means of active controllers on the chamber walls—control is to be implemented on the boundary only. We give here concise results of exact controllability for this system of interactions, with the control functions being applied through ∂Ω . In particular, it is seen that for special geometries, control may be exerted on the boundary segment Γ0 only. Moreover, this reachability problem is posed and solved on the finite energy spaces which are “natural” to the respective wave components; namely, H1×L2. In this work, we make use of microlocal estimates derived for the Neumann-control of wave equations, as well as a special vector field which is now known to exist under certain geometrical situations.

Coupled parabolic–hyperbolic Stokes–Lamé PDE system: limit behaviour of the resolvent operator on the imaginary axis

Abstract We investigate decay properties for a system of coupled partial differential equations which model the interaction between acoustic waves in a cavity and the walls of the cavity. In this system a wave equation is coupled to a structurally damped plate or beam equation. The underlying semigroup for this system is not uniformly stable, but when the system is appropriately restricted we obtain some uniform stability. We present two results of this type. For the first result, we assume that the initial wave data is zero, and the initial plate or beam data is in the natural energy space; then the corresponding solution to system decays uniformly to zero. For the second result, we assume that the initial condition is in the natural energy space and the control function is L 2 (0,∞) (in time) into the control space; then the beam displacement and velocity are both L 2 (0,∞) into a space with two spatial derivatives.

Uniform Decays in Nonlinear Thermoelastic Systems

In this paper we consider a natural robustness question for a model for structural acoustics. This model, which has been of great interest in recent years, is represented by a wave equation in