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The cost of approximate controllability for heat equations: The linear case. (English) Zbl 1007.93034

Let us consider the system

${y}_{t}-▵y+ay=v{1}_{𝒪}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}Q={\Omega }×\left(0,T\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}y=0\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega }×\left(0,T\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}y\left(x,0\right)={y}_{0}\left(x\right)\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },$

where ${\Omega }$ is a bounded domain of ${ℝ}^{d}$ with ${C}^{2}$-boundary $\partial {\Omega }$, $𝒪$ is a (probably small) non-empty open subset of ${\Omega }$, ${1}_{𝒪}$ denotes the characteristic function of $𝒪$, ${y}_{0}\in {L}^{2}\left({\Omega }\right)$ is fixed and $a\left(x,t\right)$ is a function in ${L}^{\infty }\left(Q\right)$. Hence, the control $v\left(x,t\right)$ is acting only on $q=𝒪×\left(0,T\right)$. 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 $exp\left(C/\epsilon \right)$. 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 $exp\left(C/\sqrt{\epsilon }\right)$ and that this estimate is sharp.

##### MSC:
 93C20 Control systems governed by PDE 93B05 Controllability 35K05 Heat equation