*(English)*Zbl 1007.93034

Let us consider the system

where ${\Omega}$ is a bounded domain of ${\mathbb{R}}^{d}$ with ${C}^{2}$-boundary $\partial {\Omega}$, $\mathcal{O}$ is a (probably small) non-empty open subset of ${\Omega}$, ${1}_{\mathcal{O}}$ denotes the characteristic function of $\mathcal{O}$, ${y}_{0}\in {L}^{2}\left({\Omega}\right)$ is fixed and $a(x,t)$ is a function in ${L}^{\infty}\left(Q\right)$. Hence, the control $v(x,t)$ is acting only on $q=\mathcal{O}\times (0,T)$. It is well known that this system is approximately controllable at any fixed time $T>0$ by taking controls $v\in {L}^{2}\left(q\right)$ in the sense that, for each $\epsilon >0$ and ${y}_{1}\in {L}^{2}\left({\Omega}\right)$, there exists $v\in {L}^{2}\left(q\right)$ such that the corresponding solution ${y}_{v}$ of the system satisfies $\parallel {y}_{v}\left(T\right)-{y}_{1}{\parallel}_{{L}^{2}\left({\Omega}\right)}<\epsilon $. Of course, there exist infinite controls $v$ satisfying this property. The objective of the paper is to obtain explicit bounds on the cost of approximate controllability for the system, i.e. the infimum of ${\parallel v\parallel}_{{L}^{2}\left(q\right)}$ over all $v$ satisfying the previous property. It is proved that this cost is of order $exp(C/\epsilon )$. The simultaneous finite-approximate controllability is also investigated. The proofs combine global Carleman estimates, energy estimates for parabolic equations and the variational approach to approximate controllability. When the coefficient $a$ is constant, a different approach is used to show that the cost is of order $exp(C/\sqrt{\epsilon})$ and that this estimate is sharp.