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The cost of approximate controllability for heat equations: The linear case. (English) Zbl 1007.93034
Let us consider the system$$ y_t - \triangle y + a y = v 1_{\cal O} \text{ in } Q = \Omega \times (0,T),\ \ y = 0\text{ on } \partial \Omega \times (0,T), \ \ y(x,0) = y_0(x)\text{ in } \Omega,$$ where $\Omega$ is a bounded domain of $\bbfR^d$ with $C^2$-boundary $\partial \Omega$, ${\cal O}$ is a (probably small) non-empty open subset of $\Omega$, $1_{\cal O}$ denotes the characteristic function of ${\cal O}$, $y_0 \in L^2(\Omega)$ is fixed and $a(x,t)$ is a function in $L^\infty(Q)$. Hence, the control $v(x,t)$ is acting only on $q = {\cal O} \times (0,T)$. It is well known that this system is approximately controllable at any fixed time $T >0$ by taking controls $ v \in L^2(q)$ in the sense that, for each $\varepsilon >0$ and $y_1 \in L^2(\Omega)$, there exists $ v \in L^2(q)$ such that the corresponding solution $y_v$ of the system satisfies $ \|y_v(T)- y_1\|_{L^2(\Omega)} < \varepsilon $. 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 $\|v\|_{L^2(q)}$ over all $v$ satisfying the previous property. It is proved that this cost is of order $\exp{(C/ \varepsilon)}$. 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 $\exp{(C/ \sqrt{\varepsilon})}$ and that this estimate is sharp.

93C20Control systems governed by PDE
35K05Heat equation