In this paper, we solve a local state observation problem for stochastic hyperbolic equations without boundary conditions, which is reduced to a local unique continuation property for these equations. This result is proved by a global Carleman estimate. As far as we know, this is the first result in this topic.
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DOI : 10.1051/cocv/2019049
Mots-clés : Stochastic hyperbolic equation, Carleman estimate, local state observation, local unique continuation
@article{COCV_2020__26_1_A79_0,
author = {L\"u, Qi and Yin, Zhongqi},
title = {Local state observation for stochastic hyperbolic equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {26},
year = {2020},
doi = {10.1051/cocv/2019049},
mrnumber = {4156827},
zbl = {1460.93018},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2019049/}
}
TY - JOUR AU - Lü, Qi AU - Yin, Zhongqi TI - Local state observation for stochastic hyperbolic equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2019049/ DO - 10.1051/cocv/2019049 LA - en ID - COCV_2020__26_1_A79_0 ER -
%0 Journal Article %A Lü, Qi %A Yin, Zhongqi %T Local state observation for stochastic hyperbolic equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2019049/ %R 10.1051/cocv/2019049 %G en %F COCV_2020__26_1_A79_0
Lü, Qi; Yin, Zhongqi. Local state observation for stochastic hyperbolic equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 79. doi : 10.1051/cocv/2019049. http://www.numdam.org/articles/10.1051/cocv/2019049/
[1] , Non-unicité du problème de Cauchy. Ann. Math. 117 (1983) 77–108. | DOI | MR | Zbl
[2] , Integral Geometry and Inverse Problems for Kinetic Equations, Doctoral dissertation. Novosivbirsk (1988).
[3] , and , Carleman estimate and cotrollability of linear stochastic heat equations. Appl. Math. Optim. 47 (2003) 97–120. | DOI | MR | Zbl
[4] , Sur un problème d’unicité pour les systèmes d’ équations aux derivées partielles à deux variables independantes. Ark. Mat. Astr. Fys. 26B 17 (1939) 1–9. | MR | Zbl
[5] , Uniqueness in the Cauchy problem for partial differential equations. Am. J. Math. 80 (1958) 16–36. | DOI | MR | Zbl
[6] , On the uniqueness of the Cauchy problem under partial analyticity assumptions. Geometrical optics and related topics (Cortona, 1996). Vol. 32 of Progr. Nonlinear Differential Equations Appl. Birkhäuser, Boston, MA (1997) 179–219. | MR | Zbl
[7] , The analysis of linear partial differential operators IV. Fourier integral operators. Classics in Mathematics. Springer-Verlag, Berlin (2009). | DOI | MR | Zbl
[8] , Carleman estimates, uniform Sobolev inequalities for second-order differential operators, and unique continuation theorems. Vols. 1 and 2 of Proceedings of the International Congress of Mathematicians. Amer. Math. Soc., Providence, RI (1987) 948–960. | MR | Zbl
[9] , and , Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. Duke Math. J. 55 (1987) 329–347. | DOI | MR | Zbl
[10] and , A quantitative boundary unique continuation for stochastic parabolic equations. J. Math. Anal. Appl. 402 (2013) 518–526. | DOI | MR | Zbl
[11] , Global Carleman estimate for stochastic parabolic equations, and its application. ESAIM: COCV 20 (2014) 823–839. | Numdam | MR | Zbl
[12] , Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems. Inverse Prob. 28 (2012) 045008. | DOI | MR | Zbl
[13] , Observability estimate for stochastic Schrödinger equations and its applications. SIAM J. Control Optim. 51 (2013) 121–144. | DOI | MR | Zbl
[14] , Observability estimate and state observation problems for stochastic hyperbolic equations. Inverse Probl. 29 (2013) 095011. | DOI | MR | Zbl
[15] and , Unique continuation for stochastic heat equations. ESAIM: COCV 21 (2015) 378–398. | Numdam | MR | Zbl
[16] and , Global uniqueness for an inverse stochastic hyperbolic problem with three unknowns. Commun. Pure Appl. Math. 68 (2015) 948–963. | DOI | MR | Zbl
[17] , Théorème d’unicité adapté au contrôle des solutions des problèmes hyperboliques. Commun. Part. Differ. Equ. 16 (1991) 789–800. | DOI | MR | Zbl
[18] and , Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients. Invent. Math. 131 (1998) 493–539. | DOI | MR | Zbl
[19] , Uniqueness in Cauchy problems for hyperbolic differential operators. Trans. Am. Math. Soc. 333 (1992) 821–833. | DOI | MR | Zbl
[20] and , Null controllability for forward and backward stochastic parabolic equations. SIAM J. Control Optim. 48 (2009) 2191–2216. | DOI | MR | Zbl
[21] , Unique continuation for solutions to PDE’s; between Hörmander’s theorem and Holmgren’s theorem. Commun. Part. Differ. Equ. 20 (1995) 855–884. | DOI | MR | Zbl
[22] , Unique continuation problems for partial differential equations. Geometric methods in inverse problems and PDE control, Vol. 137 of IMA Vol. Math. Appl. Springer, New York (2004) 239–255. | MR | Zbl
[23] , Carleman estimates, unique continuation and applications. Available from: http://www.math.berkeley.edu/~tataru/papers/ucpnotes.ps.
[24] , Unique continuation for stochastic parabolic equations. Differ. Integr. Equ. 21 (2008) 81–93. | MR | Zbl
[25] , Carleman and observability estimates for stochastic wave equations. SIAM J. Math. Anal. 40 (2008) 851–868. | DOI | MR | Zbl
[26] , A unified controllability/observability theory for some stochastic and deterministic partial differential equations. Proceedings of the International Congress of Mathematicians. Vol. IV. Hindustan Book Agency, New Delhi (2010) 3008–3034. | MR | Zbl
[27] , Uniqueness and nonuniqueness in the Cauchy problem. Vol. 33 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA (1983). | DOI | MR | Zbl
Cité par Sources :
This work is partially supported by NSF of China under grants 11971334 and 11931011, and the General Fund Project of Sichuan Provincial Department of Education in China under grant 13ZB0164.
