no. 4
An internal observability estimate for stochastic hyperbolic equations
Fu, Xiaoyu1 ; Liu, Xu2; Lü, Qi1 ; Zhang, Xu1
1 School of Mathematics, Sichuan University, Chengdu 610064, P.R. China.
2 School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R. China.
ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 4, pp. 1382-1411

This paper is addressed to establishing an internal observability estimate for some linear stochastic hyperbolic equations. The key is to establish a new global Carleman estimate for forward stochastic hyperbolic equations in the L 2 -space. Different from the deterministic case, a delicate analysis on the adaptedness for some stochastic processes is required in the stochastic setting.

Received:
Accepted:
DOI: 10.1051/cocv/2016042
Classification: 93B05, 93B07, 93C20
Keywords: Stochastic hyperbolic equation, observability estimate, global Carleman estimate, adaptedness, optimal control

Fu, Xiaoyu 1; Liu, Xu 2; Lü, Qi 1; Zhang, Xu 1

1 School of Mathematics, Sichuan University, Chengdu 610064, P.R. China.
2 School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R. China. 
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     title = {An internal observability estimate for stochastic hyperbolic equations},
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     pages = {1382--1411},
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Fu, Xiaoyu; Liu, Xu; Lü, Qi; Zhang, Xu. An internal observability estimate for stochastic hyperbolic equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 4, pp. 1382-1411. doi: 10.1051/cocv/2016042

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilizion of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024–1065. | Zbl | DOI

J.-M. Coron, Control and Nonlinearity. Vol. 136 of Mathematical Surveys and Monographs. American Mathematical Society Providence, RI (2007). | Zbl

T. Duyckaerts, X. Zhang and E. Zuazua, On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials. Ann. Inst. Henri Poincaré, Anal. Non Lin. 25 (2008) 1–41. | Zbl | Numdam | DOI

X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations. SIAM J. Control Optim. 46 (2007) 1578–1614. | Zbl | DOI

L. Hörmander, Linear Partial Differential Operators. Springer-Verlag, Berlin (1963). | Zbl

O.Yu. Imanuvilov, On Carleman estimates for hyperbolic equations. Asymptot. Anal. 32 (2002) 185–220. | Zbl

M.V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems. J. Inverse Ill-Posed Probl. 21 (2013) 477–560. | Zbl | DOI

M.M. Lavrent’ev, V.G. Romanov and S.P. Shishat·skiĭ, Ill-Posed Problems of Mathematical Physics and Analysis. Vol. 64 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI (1986). | Zbl

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev. 30 (1988) 1–68. | Zbl | MR | DOI

X. Liu, Global Carleman estimate for stochastic parabolic equaitons, and its application. ESAIM: COCV 20 (2014) 823–839. | Zbl | Numdam

Y. Liu, Some sufficient conditions for the controllability of wave equations with variable coefficients. Acta Appl. Math. 128 (2013) 181–191. | Zbl | DOI

Q. Lü, Observability estimate and state observation problems for stochastic hyperbolic equations. Inverse Probl. 29 (2013) 095011. | Zbl | DOI

Q. Lü and X. Zhang, Global uniqueness for an inverse stochastic hyperbolic problem with three unknowns. Commun. Pure Appl. Math. 68 (2015) 948–963. | Zbl | DOI

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open problems. SIAM Rev. 20 (1978) 639–739. | Zbl | DOI

X. Zhang, Explicit observability estimate for the wave equation with potential and its application. R. Soc. Lond. Proc., Ser. A, Math. Phys. Eng. Sci. 456 (2000) 1101–1115. | Zbl | DOI

X. Zhang, Carleman and observability estimates for stochastic wave equations. SIAM J. Math. Anal. 40 (2008) 851–868. | Zbl | DOI

E. Zuazua, Exact controllability for semilinear wave equations in one space dimension. Ann. Inst. Henri Poincaré, Anal. Non Lin. 10 (1993) 109–129. | Zbl | MR | Numdam | DOI

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