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The cost of approximate controllability for heat equations: The linear case. (English) Zbl 1007.93034
Let us consider the system \[ y_t - \triangle y + a y = v 1_{\mathcal O} \text{ in } Q = \Omega \times (0,T),\;\;y = 0\text{ on } \partial \Omega \times (0,T), \;\;y(x,0) = y_0(x)\text{ in } \Omega, \] where \(\Omega\) is a bounded domain of \(\mathbb{R}^d\) 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 \({\mathcal O}\), \(y_0 \in L^2(\Omega)\) is fixed and \(a(x,t)\) is a function in \(L^\infty(Q)\). Hence, the control \(v(x,t)\) is acting only on \(q = {\mathcal O} \times (0,T)\). It is well known that this system is approximately controllable at any fixed time \(T >0\) by taking controls \( v \in L^2(q)\) 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.

93C20 Control/observation systems governed by partial differential equations
93B05 Controllability
35K05 Heat equation