Localization of energy and localized controllability

Abstract

We will consider both the controlled Schr\"odinger equation and the controlled wave equation on a bounded open set $\Omega$ of $\RR^{N}$ during an interval of time $(0,T)$, with $T>0$. The control is distributed and acts on a nonempty open subdomain $\omega$ of $\Omega$. On the other hand, we consider another open subdomain $D$ of $\Omega$ and the localized energy of the solution in $D$. The first question we want to study is the possibility of obtaining a prescribed value of this local energy at time $T$ by choosing the control adequately. It turns out that this question is equivalent to a problem of exact or approximate controllability in $D$, which we call localized controllability and which is the second question studied in this article. We obtain sharp results on these two questions and, of course, the answers will require conditions on $\omega$ and $T$ which will be given precisely later on.