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 ${\mathbb{R}}_{+}$ or ${\mathbb{R}}^{N}$. 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.

Keywords: null controllability, weighted observability inequalities

@article{COCV_2004__10_3_381_0, author = {Cannarsa, Piermarco and Martinez, Patrick and Vancostenoble, Judith}, title = {Null controllability of the heat equation in unbounded domains by a finite measure control region}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {381--408}, publisher = {EDP-Sciences}, volume = {10}, number = {3}, year = {2004}, doi = {10.1051/cocv:2004010}, mrnumber = {2084329}, zbl = {1091.93011}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2004010/} }

TY - JOUR AU - Cannarsa, Piermarco AU - Martinez, Patrick AU - Vancostenoble, Judith TI - Null controllability of the heat equation in unbounded domains by a finite measure control region JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2004 SP - 381 EP - 408 VL - 10 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2004010/ DO - 10.1051/cocv:2004010 LA - en ID - COCV_2004__10_3_381_0 ER -

%0 Journal Article %A Cannarsa, Piermarco %A Martinez, Patrick %A Vancostenoble, Judith %T Null controllability of the heat equation in unbounded domains by a finite measure control region %J ESAIM: Control, Optimisation and Calculus of Variations %D 2004 %P 381-408 %V 10 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2004010/ %R 10.1051/cocv:2004010 %G en %F COCV_2004__10_3_381_0

Cannarsa, Piermarco; Martinez, Patrick; Vancostenoble, Judith. Null controllability of the heat equation in unbounded domains by a finite measure control region. ESAIM: Control, Optimisation and Calculus of Variations, Volume 10 (2004) no. 3, pp. 381-408. doi : 10.1051/cocv:2004010. http://archive.numdam.org/articles/10.1051/cocv:2004010/

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