Exact Internal Controllability for a Problem with Imperfect Interface

TL;DR

This paper establishes the internal exact controllability of a second order linear evolution equation in a two-component domain with an imperfect interface, using observability inequalities and the Hilbert Uniqueness Method.

Contribution

It introduces a novel controllability approach for problems with imperfect interfaces, incorporating jump conditions and geometric considerations.

Findings

Derived an observability inequality for the system.

Constructed exact controls via the Hilbert Uniqueness Method.

Identified a lower bound for the control time based on domain geometry and interface properties.

Abstract

In this paper we study the internal exact controllability for a second order linear evolution equation defined in a two-component domain. On the interface we prescribe a jump of the solution proportional to the conormal derivatives, meanwhile a homogeneous Dirichlet condition is imposed on the exterior boundary. Due to the geometry of the domain, we apply controls through two regions which are neighborhoods of a part of the external boundary and of the whole interface, respectively. Our approach to internal exact controllability consists in proving an observability inequality by using the Lagrange multipliers method. Eventually we apply the Hilbert Uniqueness Method, introduced by J.-L. Lions, which leads to the construction of the exact control through the solution of an adjoint problem. Finally we find a lower bound for the control time depending not only on the geometry of our domain and on the matrix of coefficients of our problem but also on the coefficient of proportionality of the jump with respect to the conormal derivatives.