Exact null internal controllability for the heat equation on unbounded convex domains∗. (6th February 2014)

• Record Type: Journal Article
• Authors: Barbu, Viorel
• Is Part Of: ESAIM. Volume 20:Number 1(2014:Jan.)
• Page Start: 222
• Page End: 235
• Publication Date: 2014-02-06
• Subjects: System analysis -- Periodicals; Calculus of variations -- Periodicals; Mathematical analysis -- Periodicals; Mathematical optimization -- Periodicals; Control theory -- Periodicals; 515.64
• DOI: 10.1051/cocv/2013062
• Languages: English
• ISSNs: 1292-8119

Abstract

<abstract abstract-type="normal" xml:lang="en"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>The liner parabolic equation <inline-formula><alternatives><tex-math id="tex_eq1"><![CDATA[\hbox{$\frac{\pp y}{\pp t}-\frac12\, \D y+F\cdot\na y={\vec{1}}_{\calo_0}u$}]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2zjwwzqh" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math id="mml_eq1" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mfrac><mml:mi mathvariant="italic">∂y</mml:mi><mml:mi mathvariant="italic">∂t</mml:mi></mml:mfrac></mml:mrow><mml:mo mathvariant="normal">−</mml:mo><mml:mrow><mml:mfrac><mml:mn mathvariant="normal">1</mml:mn><mml:mn mathvariant="normal">2</mml:mn></mml:mfrac></mml:mrow><mml:mo> </mml:mo><mml:mi mathvariant="italic">Δy</mml:mi><mml:mo mathvariant="normal">+</mml:mo><mml:mi mathvariant="italic">F</mml:mi><mml:mo mathvariant="normal">·</mml:mo><mml:mi mathvariant="normal">∇</mml:mi><mml:mi mathvariant="italic">y</mml:mi><mml:mo mathvariant="normal">=</mml:mo><mml:msub><mml:mrow><mml:mn mathvariant="bold">1</mml:mn></mml:mrow><mml:msub><mml:mi mathvariant="normal">𝒪</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub></mml:msub><mml:mi mathvariant="italic">u</mml:mi></mml:math></alternatives></inline-formula> with Neumann boundary condition on a convex open domain 𝒪 ⊂ ℝ<sup><italic>d</italic></sup> with smooth boundary is exactly null controllable on each finite interval if 𝒪<sub>0</sub> is an open subset of 𝒪 which contains a suitable neighbourhood of the recession cone of <inline-formula><alternatives><tex-math id="tex_eq5"><![CDATA[\hbox{$\ov\calo$}]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2zjwwzsm" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math id="mml_eq5" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:menclose notation="top"><mml:mi mathvariant="normal">𝒪</mml:mi></mml:menclose></mml:math></alternatives></inline-formula>. Here, <italic>F</italic> : ℝ<sup><italic>d</italic></sup> → ℝ<sup><italic>d</italic></sup> is a bounded, <italic>C</italic><sup>1</sup>-continuous function, and <italic>F</italic> = ∇<italic>g</italic>, where <italic>g</italic> is convex and coercive.</p> </abstract>