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

S Monsurrò,A K Nandakumaran,C Perugia

doi:10.1007/s11587-021-00625-w

Abstract

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.

Highlights

Let us consider an open bounded subset Ω of Rn, n ≥ 2, and let Ω1 = Ω \ Ω2, where Ω2 is an open and bounded set such that Ω2 ⊂⊂ Ω
The discontinuity of the solution is the mathematical interpretation of imperfect interface
The boundary exact controllability problem consists in finding a suitable control, acting on the external boundary or even on just a part of it, driving the trajectories of an evolution system to a desired state at a certain time T > 0, for all initial data

Summary

Introduction

In the above mentioned domain we want to study the boundary exact controllability problem for a hyperbolic system of equations with appropriate boundary and interface conditions on ∂Ω and on Γ. The boundary exact controllability problem consists in finding a suitable control, acting on the external boundary or even on just a part of it, driving the trajectories of an evolution system to a desired state at a certain time T > 0, for all initial data It reduces to prove an estimate for the energy of an uncontrolled system, at time t = 0, through partial measurements of its solution done on the boundary control set. Lions studies for the first time the exact controllability, via HUM, for the wave equation with transmission conditions He considers a Dirichlet problem with a matrix constant on each component of the domain and a control set on part of the external boundary. For what concerns the internal exact controllability of hyperbolic problems in composites with imperfect interface we refer to Faella et al [14], Monsurrò and Perugia [29] and Monsurrò et al [28]

Setting of the problem

The observability inequality

The boundary exact controllability result