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.

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.

DOI : 10.1051/cocv/2013085
Classification : 93B05, 93B07
Mots-clés : Carleman estimate, stochastic parabolic equation, insensitizing control, controllability 
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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. http://www.numdam.org/articles/10.1051/cocv/2013085/

[1] F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. De Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey. Math. Control Relat. Fields 1 (2011) 267-306. | MR | Zbl

[2] O. Bodart, M. González-Burgos and R. Pérez-Garcia, A local result on insensitizing controls for a semilinear heat equation with nonlinear boundary Fourier conditions. SIAM J. Control Optim. 43 (2004) 955-969. | MR | Zbl

[3] A.-P. Calderón, Uniqueness in the Cauchy problem for partial differential equations. Amer. J. Math. 80 (1958) 16-36. | MR | Zbl

[4] T. Carleman, Sur un problème d'unicité pur les systèmes d'équations aux dérivées partielles à deux variables indépendantes. Ark. Mat. Astr. Fys. 26 (1939) 1-9. | MR | Zbl

[5] X. Fu, A weighted identity for partial differential operator of second order and its applications. C.R. Math. Acad. Sci. Paris 342 (2006) 579-584. | MR

[6] A.V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations. In vol. 34, Lect. Notes Ser. Seoul National University, Seoul, Korea (1996). | MR | Zbl

[7] L. Hörmander, Linear Partial Differential Operators, in vol. 116. Die Grundlehren der mathematischen Wissenschaften. Academic Press, New York (1963). | Zbl

[8] O. Yu. Imanuvilov and J.P. Puel, Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems. Int. Math. Res. Not. 16 (2003) 883-913. | MR | Zbl

[9] O. Yu. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations. Publ. Res. Inst. Math. Sci. 39 (2003) 227-274. | MR | Zbl

[10] N.V. Krylov, A Wn2-theory of the Dirichlet problem for SPDEs in general smooth domains. Prob. Theory Related Fields 98 (1994) 389-421. | MR | Zbl

[11] X. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems. Birkhäuser Boston, Inc., Boston (1995). | MR | Zbl

[12] J.-L. Lions, Quelques notions dans l'analyse et le contrôle de systèmes à données incomplètes, Proc. of the XIth Congress on Differential Equations and Applications/First Congress on Appl. Math. Univ. Málaga, Málaga (1990) 43-54. | MR | Zbl

[13] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, vol. 170. Springer-Verlag, New York-Berlin (1971). | MR | Zbl

[14] X. Liu, Insensitizing controls for a class of quasilinear parabolic equations. J. Differ. Eqs. 253 (2012) 1287-1316. | MR | Zbl

[15] X. Liu and X. Zhang, Local controllability of multidimensional quasi-linear parabolic equations. SIAM J. Control Optim. 50 (2012) 2046-2064. | MR | Zbl

[16] Q. Lü, Some results on the controllability of forward stochastic heat equations with control on the drift. J. Funct. Anal. 260 (2011) 832-851. | MR | Zbl

[17] Q. Lü, Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems. Inverse Problems 28 (2012) 045008. | MR | Zbl

[18] Q. Lü and X. Zhang, Carleman estimates for parabolic operators with discontinuous and anisotropic diffusion coefficients, an elementary approach. In preparation.

[19] Q. Lü and X. Zhang, General Pontryagin-type stochastic maximum principle and backward stochastic evolution equations in infinite dimensions, arXiv:1204.3275.

[20] J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications. Stochastic Process. Appl. 70 (1997) 59-84. | MR | Zbl

[21] E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes. Stochastic 3 (1979) 127-167. | MR | Zbl

[22] J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations. J. Differ. Eqs. 66 (1987) 118-139. | MR | Zbl

[23] S. Tang and X. Zhang, Null controllability for forward and backward stochastic parabolic equations. SIAM J. Control Optim. 48 (2009) 2191-2216. | MR | Zbl

[24] L. De Teresa, Insensitizing controls for a semilinear heat equation. Commun. Partial Differ. Eqs. 25 (2000) 39-72. | MR | Zbl

[25] Y. Yan and F. Sun, Insensitizing controls for a forward stochastic heat equation. J. Math. Anal. Appl. 384 (2011) 138-150. | MR | Zbl

[26] X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations. Proc. of the Int. Congress of Math., Vol. IV. Hyderabad, India (2010) 3008-3034. | MR | Zbl

[27] X. Zhou, A duality analysis on stochastic partial differential equations. J. Funct. Anal. 103 (1992) 275-293. | MR | Zbl

[28] X. Zhou, On the necessary conditions of optimal controls for stochastic partial differential equations. SIAM J. Control Optim. 31 (1993) 1462-1478. | MR | Zbl

[29] E. Zuazua, Controllability and Observability of Partial Differential Equations: Some results and open problems. Handbook of Differential Equations: Evol. Differ. Eqs., vol. 3. Elsevier Science (2006) 527-621. | MR | Zbl

[30] C. Zuily, Uniqueness and Non-Uniqueness in the Cauchy Problem. Birkhäuser Verlag, Boston-Basel-Stuttgart (1983). | MR | Zbl

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