Unique continuation for stochastic heat equations

Lü, Qi; Yin, Zhongqi

doi:10.1051/cocv/2014027

Lü, Qi ¹ ; Yin, Zhongqi ²

¹ School of Mathematics, Sichuan University, Chengdu 610064, P.R. China.
² School of Mathematics, Sichuan Normal University, Chengdu 610068, P.R. China.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 378-398.

Résumé

We establish a unique continuation property for stochastic heat equations evolving in a domain $G \subset R^{n}$ ( $n \in N$ ). Our result shows that the value of the solution can be determined uniquely by means of its value on an arbitrary open subdomain of $G$ at any given positive time constant. Further, when $G$ is convex and bounded, we also give a quantitative version of the unique continuation property. As applications, we get an observability estimate for stochastic heat equations, an approximate result and a null controllability result for a backward stochastic heat equation.

Reçu le : 2013-10-21

MR Zbl | 2 citations dans Numdam

DOI : 10.1051/cocv/2014027

Classification : 60H15, 93B05
Mots-clés : Stochastic heat equations, unique continuation property, backward stochastic heat equations, approximate controllability, null controllability

Affiliations des auteurs :

Lü, Qi ¹ ; Yin, Zhongqi ²

¹ School of Mathematics, Sichuan University, Chengdu 610064, P.R. China.
² School of Mathematics, Sichuan Normal University, Chengdu 610068, P.R. China.

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     author = {L\"u, Qi and Yin, Zhongqi},
     title = {Unique continuation for stochastic heat equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {378--398},
     publisher = {EDP-Sciences},
     volume = {21},
     number = {2},
     year = {2015},
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     mrnumber = {3348404},
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     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv/2014027/}
}

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Lü, Qi; Yin, Zhongqi. Unique continuation for stochastic heat equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 378-398. doi : 10.1051/cocv/2014027. https://www.numdam.org/articles/10.1051/cocv/2014027/

Bibliographie
Cité par

J.P. Dauer, N.I. Mahmudov and M.M. Matar, Approximate controllability of backward stochastic evolution equations in Hilbert spaces. J. Math. Anal. Appl. 323 (2006) 42–56. | DOI | MR | Zbl

L. De Teresa, Approximate controllability of a semilinear heat equation in $𝐑^{N}$ . SIAM J. Control Optim. 36 (1998) 2128–2147. | DOI | MR | Zbl

L. Escauriaza, Carleman inequalities and the heat operator. Duke Math. J. 104 (2000) 113–127. | DOI | MR | Zbl

L. Escauriaza and L. Vega, Carleman inequalities and the heat operator II. Indiana Univ. Math. J. 50 (2001) 1149–1169. | DOI | MR | Zbl

C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 31–61. | DOI | MR | Zbl

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J. Control Optim. 45 (2006) 1399–1446. | DOI | MR | Zbl

E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: the linear case. Adv. Differ. Equ. 5 (2000) 465–514. | MR | Zbl

E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. Henri Poincaré Anal., Non Linéaire 17 (2000) 583–616. | DOI | Numdam | MR | Zbl

E. Fernández-Cara, M.J. Garrido-Atienza and J. Real, On the approximate controllability of a stochastic parabolic equation with a multiplicative noise. C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 675–680. | DOI | MR | Zbl

A.V. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations. Vol. 34 of Lect. Notes Ser. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996). | MR | Zbl

J.Hadamard, Lectures on Cauchy’s problem in linear partial differential equations. Dover Publications, New York (1953). | MR | Zbl

L. Hörmander, The Analysis of Linear Partial differential operators. III. Pseudodifferential operators. Vol. 274 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin (1985). | MR | Zbl

M.M. Lavrentév, V.G. Romanov and S.P. Shishat-skii, Ill-posed Problems of Mathematical Physics and Analysis, Translated from the Russian by J.R. Schulenberger. Vol. 64 of Trans. Math. Monogr. AMS, Providence, RI (1986). | MR | Zbl

G. Lebeau and L. Robbiano, Contrôle exact de l’équation de la chaleur. Commun. Partial Differ. Equ. 20 (1995) 335–356. | DOI | MR | Zbl

H. Li and Q. Lü, A quantitative boundary unique continuation for stochastic parabolic equations. J. Math. Anal. Appl. 402 (2013) 518–526. | DOI | MR | Zbl

F. Lin, A uniqueness theorem for parabolic equations. Commun. Pure Appl. Math. 43 (1990) 127–136. | DOI | MR | Zbl

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

Q. Lü, Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems. Inverse Probl. 28 (2012) 045008. | DOI | MR | Zbl

N.I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. SIAM J. Control Optim. 42 (2003) 1604–1622. | DOI | MR | Zbl

K.-D. Phung and G. Wang, Quantitative unique continuation for the semilinear heat equation in a convex domain. J. Funct. Anal. 259 (2010) 1230–1247. | DOI | MR | Zbl

K.-D. Phung and G. Wang, An observability estimate for parabolic equations from a measurable set in time and its applications. J. Eur. Math. Soc. 15 (2013) 681–703. | DOI | MR | Zbl

K.-D. Phung, L. Wang and C. Zhang, Bang-bang property for time optimal control of semilinear heat equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 31 (2014) 477–499. | DOI | Numdam | MR | Zbl

C. Poon, Unique continuation for parabolic equations. Commun. Partial Differ. Equ. 21 (1996) 521–539. | MR | Zbl

J. Real, Some results on controllability for stochastic heat and Stokes equations. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 1197–1202. | MR | Zbl

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

S. Vessella, Carleman estimates, optimal three cylinder inequality, and unique continuation properties for solutions to parabolic equations. Commun. Partial Differ. Equ. 28 (2003) 637–676. | DOI | MR | Zbl

M. Yamamoto, Carleman estimates for parabolic equations and applications. Inverse Probl. 25 (2009) 123013. | DOI | MR | Zbl

X. Zhang, Unique continuation for stochastic parabolic equations. Differ. Integral Equ. 21 (2008) 81–93. | MR | Zbl

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

E. Zuazua, Controllability and observability of partial differential equations: some results and open problems. Vol. III of Handbook of differential equations: evolutionary equations. Elsevier/North-Holland, Amsterdam (2007). | MR | Zbl

C. Zuily, Uniqueness and nonuniqueness in the Cauchy Problem. Birkhäuser Boston, Inc., Boston, MA (1983). | MR | Zbl

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