A note on the exact boundary controllability for an imperfect transmission problem

In this note, we consider a hyperbolic system of equations in a domain made up of two components. We prescribe a homogeneous Dirichlet condition on the exterior boundary and a jump of the displacement proportional to the conormal derivatives on the interface. This last condition is the mathematical interpretation of an imperfect interface. We apply a control on the external boundary and, by means of the Hilbert Uniqueness Method, introduced by J. L. Lions, we study the related boundary exact controllability problem. The key point is to derive an observability inequality by using the so called Lagrange multipliers method, and then to construct the exact control through the solution of an adjoint problem. Eventually, we prove a lower bound for the control time which depends on the geometry of the domain, on the coefficients matrix and on the proportionality between the jump of the solution and the conormal derivatives on the interface.

On the Exact Boundary Control for the Linear Klein-Gordon Equation in Non-cylindrical Domains

The purpose of this paper is to study an exact boundary controllability problem in noncylindrical domains for the linear Klein-Gordon equation. Here, we work near of the extension techniques presented By J. Lagnese in [12] which is based in the Russell’s controllability method. The control time is obtained in any time greater then the value of the diameter of the domain on which the initial data are supported. The control is square integrable and acts on whole boundary and it is given by conormal derivative associated with the above-referenced wave operator.

Exact boundary controllability and exact boundary synchronization for a coupled system of wave equations with coupled Robin boundary controls

In this paper, we consider the exact boundary controllability and the exact boundary synchronization (by groups) for a coupled system of wave quations with coupled Robin boundary controls. Owing to the difficulty coming from the lack of regularity of the solution, we confront a bigger challenge than that in the case with Dirichlet or Neumann boundary controls. In order to overcome this difficulty, we use the regularity results of solutions to the mixed problem with Neumann boundary conditions by Lasiecka and Triggiani ([6]) to get the regularity of solutions to the mixed problem with coupled Robin boundary conditions. Thus we show the exact boundary controllability of the system, and by a method of compact perturbation, we obtain the non-exact boundary controllability of the system with fewer boundary controls on some special domains. Based on this, we further study the exact boundary synchronization (by groups) for the same system, the determination of the exactly synchronizable state (by groups), as well as the necessity of the compatibility conditions of the coupling matrices.