Controllability for a class of reaction–diffusion systems: the generalized Kalman's condition

Abstract

In this Note, we study the controllability of a class of parabolic systems of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>D</mml:mi> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>+</mml:mo> <mml:mi>B</mml:mi> <mml:msub> <mml:mi>χ</mml:mi> <mml:mi>ω</mml:mi> </mml:msub> <mml:mi>u</mml:mi> </mml:math> with Dirichlet conditions on the boundary of a bounded domain Ω , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>Ω</mml:mi> </mml:math> is a subdomain. Here <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>D</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo>∈</mml:mo> <mml:mi mathvariant="script">L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>B</mml:mi> <mml:mo>∈</mml:mo> <mml:mi mathvariant="script">L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:mo>;</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:math> and we prove that the algebraic Kalman condition extends to such systems.