An internal observability estimate for stochastic hyperbolic equations∗
1 School of Mathematics, Sichuan University, Chengdu 610064,
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2 School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R. China.
Accepted: 7 June 2016
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 L2-space. Different from the deterministic case, a delicate analysis on the adaptedness for some stochastic processes is required in the stochastic setting.
Mathematics Subject Classification: 93B05 / 93B07 / 93C20
Key words: Stochastic hyperbolic equation / observability estimate / global Carleman estimate / adaptedness / optimal control
This work is partially supported by the NSF of China under grants 11231007, 11322110, 11371084 and 11471231, by the Fundamental Research Funds for the Central Universities under grants 2412015BJ011 and 2015SCU04A02, by the Foundation for the Author of National Excellent Doctoral Dissertation of China under grant 201213, by the Program for New Century Excellent Talents in University under grant NCET-12-0812, by the Fok Ying Tong Education Foundation under grant 141001, and by the Chang Jiang Scholars Program from Chinese Education Ministry. The authors highly appreciate the anonymous referees for their constructive comments which led to this improved version.
© EDP Sciences, SMAI 2016