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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.
@article{COCV_2014__20_3_823_0,
author = {Liu, Xu},
title = {Global {Carleman} estimate for stochastic parabolic equations, and its application},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {823--839},
publisher = {EDP-Sciences},
volume = {20},
number = {3},
year = {2014},
doi = {10.1051/cocv/2013085},
mrnumber = {3264225},
zbl = {1292.93032},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013085/}
}
TY - JOUR AU - Liu, Xu TI - Global Carleman estimate for stochastic parabolic equations, and its application JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 823 EP - 839 VL - 20 IS - 3 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013085/ DO - 10.1051/cocv/2013085 LA - en ID - COCV_2014__20_3_823_0 ER -
%0 Journal Article %A Liu, Xu %T Global Carleman estimate for stochastic parabolic equations, and its application %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 823-839 %V 20 %N 3 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013085/ %R 10.1051/cocv/2013085 %G en %F COCV_2014__20_3_823_0
Liu, Xu. Global Carleman estimate for stochastic parabolic equations, and its application. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 3, pp. 823-839. doi: 10.1051/cocv/2013085
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