Differential riccati equation for the active control of a problem in structural acoustics

Abstract

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.

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

“…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.…”

“…The case α = 1; hence D 0 = 0; β = 0, in (1.2c)- (1.2d) below (see [1,3,16]). In the aforementioned references, the original model consisted of a coupled system of two partial differential equations (PDEs) of different type-a hyperbolic PDE and a parabolic-like PDE-which are strongly coupled at the interface (represented by the flexible wall Γ 0 ).…”

“…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.…”

“…In the aforementioned references, the original model consisted of a coupled system of two partial differential equations (PDEs) of different type-a hyperbolic PDE and a parabolic-like PDE-which are strongly coupled at the interface (represented by the flexible wall Γ 0 ). More precisely, the original model in the analysis of [1,3,16] is problem (1.1) below with constant α = 1 in (1.2d) and, hence, damping operator D 0 taken to be null: D 0 = 0, while the constant β is taken equal to zero: β = 0 in (1.2c). In the resulting specialization of model (1.2), a wave equation in z (a hyperbolic PDE, problems (1.2a), (1.2b), and (1.2c)) which is active within the chamber, influences through its velocity trace z t | Γ 0 on the flexible wall Γ 0 the dynamics of a plate-like abstract Euler-Bernoulli equation (1.2d) in v defined on Γ 0 and subject to (strong) Kelvin-Voight damping (a parabolic PDE with ρ > 0, α = 1).…”

Uniform stabilization of a coupled structural acoustic system byboundary dissipation

“…The results existing in the literature on the stabilization of structural acoustic models refer to those where the floor is strongly damped by means of structural damping (see [1], [2], [7]). In the case of structural damping present in the model, the component of the uncoupled system corresponding to the Euler-Bernoulli equation represents an analytic semigroup ( [22], [6]).…”