Unique continuation for stochastic heat equations

Lü, Qi; Yin, Zhongqi

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

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.

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

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