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Global Carleman estimate for stochastic parabolic equations, and its application∗

Published online by Cambridge University Press:  05 June 2014

Xu Liu
Xu Liu*
Affiliation:
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R. China. liuxu@amss.ac.cn
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Abstract

This paper is addressed to proving a new Carleman estimate for stochastic parabolic equations. Compared to the existing Carleman estimate in this respect (see [S. Tang and X. Zhang, SIAM J. Control Optim. 48 (2009) 2191–2216.], Thm. 5.2), one extra gradient term involving in that estimate is eliminated. Also, our improved Carleman estimate is established by virtue of the known Carleman estimate for deterministic parabolic equations. As its application, we prove the existence of insensitizing controls for backward stochastic parabolic equations. As usual, this insensitizing control problem can be reduced to a partial controllability problem for a suitable cascade system governed by a backward and a forward stochastic parabolic equation. In order to solve the latter controllability problem, we need to use our improved Carleman estimate to establish a suitable observability inequality for some linear cascade stochastic parabolic system, while the known Carleman estimate for forward stochastic parabolic equations seems not enough to derive the desired inequality.

Keywords

Carleman estimatestochastic parabolic equationinsensitizing controlcontrollability
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 3 , July 2014 , pp. 823 - 839
Copyright
© EDP Sciences, SMAI 2014

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