Approximate controllability for an integro-differential control problem

Abstract

We study a controlled integro-differential parabolic problem with the integral part being given by a generalized convolution and the control acting on a subdomain. We prove the approximate controllability and the basic tool is a unique continuation result at initial time, for parabolic equations which, in turn, relies on fine Carleman estimates.

Keywords:

• approximate controllability
• integro-differential parabolic equations
• unique continuation
• Carleman estimates

AMS Subject Classifications::

• Primary 35K10
• Secondary 47G20
• 93B05
• 93B07
• 93C20

Acknowledgements

The authors thank the referees for their useful remarks and suggestions that have improved this article. Cătălin Lefter was supported by CNCSIS grant ID_404/2007 – 2010 and Dipartimento di Matematica of the Università degli Studi di Milano. Alfredo Lorenzi is a member of G.N.A.M.P.A. of the Italian Istituto Nazionale di Alta Matematica. This article was partially supported by the project PRIN 2008 ‘Analisi Matematica nei Problemi Inversi per le Applicazioni’ of the Italian Ministero dell'Istruzione, dell'Università e della Ricerca.