Null controllability of the heat equation in unbounded domains by a finite measure control region

Cannarsa, Piermarco; Martinez, Patrick; Vancostenoble, Judith

ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 3, pp. 381-408.

Motivated by two recent works of Micu and Zuazua and Cabanillas, De Menezes and Zuazua, we study the null controllability of the heat equation in unbounded domains, typically ${ℝ}_{+}$ or ${ℝ}^N$. Considering an unbounded and disconnected control region of the form $\omega :={\cup }_n{\omega }_n$, we prove two null controllability results: under some technical assumption on the control parts ${\omega }_n$, we prove that every initial datum in some weighted $L^2$ space can be controlled to zero by usual control functions, and every initial datum in $L^2\left(\Omega \right)$ can be controlled to zero using control functions in a weighted $L^2$ space. At last we give several examples in which the control region has a finite measure and our null controllability results apply.

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