On exact boundary controllability for linearly coupled wave equations

Abstract

In this paper we study exact boundary controllability for a system of two linear wave equations coupled by lower order terms. We obtain square integrable control of Neuman type for initial state with finite energy, in nonsmooth domains of the plane. PreliminariesLet Ω ⊂ R 2 be a bounded simply connected domain with piecewise C ∞ boundary ∂Ω. It is assumed that ∂Ω has no cuspid points and that Ω lays in one side of ∂Ω. We will refer to this type of domain as curvilinear polygon. The exterior unit normal vector η is defined almost everywhere in ∂Ω. Points of R 2 × R are denoted by (x 1 , x 2 , t), denotes the Laplacian operator in the spatial variable x = (x 1 , x 2 ) ∈ R 2 and subscripts are used for partial derivatives. Consider the systemwith T > 0, α, β, γ , δ real constants and ∂Ω = Γ 0 ∪ Γ 1 is a decomposition of the boundary where the parts Γ 0 and Γ 1 have disjoint interiors but their closure may have common isolated points. Many problems in structural dynamics are modeled by this type of system. For instance, the equations (1.2) describe the transverse free vibrations of an elastically connected double-membrane compound system. In this case u and v are the displacements of two vibrating membranes measured from their equilibrium positions. The coupling terms ±α(u − v) represent forces imposed by the coupling elastic layer. If the membranes are held fixed on the portion Γ 0 of the * Corresponding author.

On the local energy decay of the synchronizable solutions of a coupled wave equations system

“…We say that the energy decay locally when there exist a real function $$$ t\hookrightarrow p(t) $$$ such that $$$$ E\left(U,\Omega, t\right)=p(t)E\left(U,\Omega, 0\right) $$$$ with $$$ p(t)\to 0 $$$ when $$$ t\to +\infty $$$. The energy decay property has important applications in many branches of theory of hyperbolic equations, in special in the in the aspects of the controllability theory for coupled wave systems (see previous works [1, 2, 14–16]). Because of the importance of the energy decay subject, we have decided to study it in the aspect of the synchronization solution.…”

Optimal polynomial decay for a coupled system of wave with past history

“…Boundary damping is also considered in [5,6]. On exact boundary controllability for linearly coupled wave equations, we refer [7]. Uniform exponential stability was given in [8] for wave equations coupled in parallel with coupling distributed springs and viscous dampers due to different boundary conditions and wave propagation speeds.…”

Energy decay and control for a system of strings elastically connected in parallel

“…In the literature, there are some papers concerning the control and stabilization problems for systems of coupled wave equations (see Bastos et al, Najafi, and Rajaram & Najafi), with the most diverse type of coupling and boundary conditions, in space dimension greater than or equal to 2. In Rajaram and Najafi, the method HUM presented in Lions, was used to obtain the desirable control results.…”

“…In Rajaram and Najafi, the method HUM presented in Lions, was used to obtain the desirable control results. Here, as in Bastos et al and Nunes and Bastos, we use the controllability method presented by D.L. Russell .…”