Avalos, George; Lasiecka, Irena Exact controllability of finite energy states for an acoustic wave/plate interaction under the influence of boundary and localized controls. (English) Zbl 1100.60504 Adv. Differ. Equ. 10, No. 8, 901-930 (2005). In this very important and useful paper the exact controllability problem associated with the controlled partial differential equation model (comprised of a three-dimensional interior acoustic wave equation coupled to a two-dimensional Kirchhoff plate equation, with the coupling being accomplished across a boundary interface). \[ z_{tt}=\Delta z \text{ on } (0,T) \times \Omega \]\[ \partial z/\partial \nu=u_1 \text{ on } (0,T) \times \Gamma_1, \partial z/\partial \nu=v_t \text{ on } (0,T) \times \Gamma_0 \]\[ v_{tt} - \gamma \Delta v_{tt}+\Delta^2v+z_t| _{\Gamma _0}=a(x)u_0 \text{ on } (0,T) \times \Gamma_0 \]\[ v| _{\partial \Gamma _0}=\Delta v| _{\partial \Gamma _0} = 0 \text{ on } (0,T) \times \partial \Gamma_0 \]\[ [z(0), z_t(0), v(0), v_t(0)]=[\mathbf{z}_0, \mathbf{v}_0], \] where functions \(u_i\) are the controls of the system, is analysed. The derivation of exact controllability properties constitutes a particularly challenging problem for coupled structures. This circumstance is due to the fact that the solution of such a problem tends to depend upon the existence of an inverse-type of estimate (the observability estimate) which allows for a reconstruction of the total energy from overdetermined boundary data. The objective of this paper is to find controls \([u_1, u_o]\) which will steer, exactly, the given (finite energy) initial data to an arbitrary target state, at time \(T>0\). Here are invoking the classical definition of exact controllability. The PDE model (1) is a representative of the class of hybrid structures (is an example of a structural acoustic interaction when it comprises to ana acoustic wave equation on the interior of \(\Omega\), coupled to a structural plate equation on the flat boundary portion). Main result: The authors show for PDE system (1) that by means of boundary controls, the interior wave and Kirchhoff plate initial data can be steered to an arbitrary finite energy state. Reconstruction/observability estimates in negative (dual) norms, with applications to the boundary controllability of the Kirchhoff plate component (1), (which is driven by a source resulting from the coupling with the boundary trace of the interior wave velocity) is presented. Sharp trace regularity theory, corresponding to the wave equation with Neumann (non-Lopatinski) boundary conditions and equally Hidden regularity of normal derivatives on the boundary in negative norms for Kirchhoff plates, under Dirichlet boundary data are proposed. Reviewer: Jan Lovíšek (Bratislava) Cited in 10 Documents MSC: 60H25 Random operators and equations (aspects of stochastic analysis) 47H10 Fixed-point theorems 34D35 Stability of manifolds of solutions to ordinary differential equations Keywords:Exact controllability; finite energy states; acoustic wave; hybrid partial differential equation; Kirchhoff plate; interaction; boundary controls; initial data; observability estimates; dual norms; sharp trace regularity; Hidden regularity negative scale; Sobolev spaces; Neumann; Dirichlet boundary conditions; inverse-type of estimate PDFBibTeX XMLCite \textit{G. Avalos} and \textit{I. Lasiecka}, Adv. Differ. Equ. 10, No. 8, 901--930 (2005; Zbl 1100.60504)