## 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.

### MSC:

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