Dr. Carmen Perugia “Some exact controllability results for an imperfect interface problem”

We study the exact boundary and internal controllability for an imperfect trans-
mission hyperbolic problem defined in a two-component domain and characterized by a jump of the solution on the interface which is proportional to the conormal
derivatives. This last condition is the mathematical interpretation of an imperfect
interface. Regarding as the exact boundary controllability result, we prescribe a non homogeneous Dirichlet condition on the external boundary, which acts as the control. For what concern the internal controllability issue, we prescribe a homogeneous Dirichlet condition on the external boundary and, due to the geometry of the domain, we apply controls through two regions which are neighborhoods of a part of the ex ternal boundary and of the whole interface, respectively. These results are achieved by performing the Hilbert Uniqueness Method (HUM for short), introduced by J. -L. Lions, which is a constructive method building the exact control through the solution of a hyperbolic problem associated to suitable initial conditions. These initial conditions are obtained by calculating at zero time the solution of an adjoint backward problem by means of a functional which turns out to be an isomorphism, thanks to an important inequality known as the observability estimate. Due to the imperfect
interface, when proving observability estimate, some difficulties arise in estimating
specific surface integrals. Moreover, unlike classical cases, we find a lower bound for
the control time T 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.