Observability estimate and state observation problems for stochastic hyperbolic equations

Lü, Qi

doi:10.1088/0266-5611/29/9/095011

Cited by 33 publications

(27 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…is a solution to a stochastic PDE and C is a boundary observation operator, inequalities in the form of (2.36) are usually called the hidden regularity of the solution, i.e., it does not follow directly from the classical trace theorem of Sobolev space. We refer the readers to [14,16,17,31] for the hidden regularity for some stochastic PDEs.…”

Section: Admissible Observation Operatormentioning

confidence: 99%

Stochastic Well-Posed Systems and Well-Posedness of Some Stochastic Partial Differential Equations with Boundary Control and Observation

Lü¹

2015

SIAM J. Control Optim.

Self Cite

View full text Add to dashboard Cite

We generalize the concept "well-posed linear system" to stochastic linear control systems and study some basic properties of such kind systems. Under our generalized definition, we show the well-posedness of the stochastic heat equation and the stochastic Schrödinger equation with suitable boundary control and observation operators, respectively.2010 Mathematics Subject Classification. Primary 93E03; Secondary 60H15.

show abstract

Section: Admissible Observation Operatormentioning

confidence: 99%

Stochastic Well-Posed Systems and Well-Posedness of Some Stochastic Partial Differential Equations with Boundary Control and Observation

Lü¹

2015

SIAM J. Control Optim.

Self Cite

View full text Add to dashboard Cite

show abstract

“…There exist numerous works devoted to observability estimates for deterministic hyperbolic equations. However, there are only a very few works addressed to similar problems but for stochastic hyperbolic equations ( [12,13,16]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In [12], by virtue of another global Carleman estimate, the result in [16] was improved to the following boundary observability inequality:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

An internal observability estimate for stochastic hyperbolic equations

Liu

Lü

et al. 2016

ESAIM: COCV

Self Cite

View full text Add to dashboard Cite

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.2010 Mathematics Subject Classification. Primary 93B05; Secondary 93B07, 93C20.

show abstract

“…In [29], a Carleman type inequality for stochastic wave equations was first obtained. The result in [29] was improved in [15] and [17] to solve some inverse problems for stochastic wave equations. In [14], the author got a Carleman type inequality for stochastic Schrödinger equations and used it to study a state observation problem for these equations.…”

Section: Introductionmentioning

confidence: 99%

Exact Controllability for Stochastic Transport Equations

Lü¹

2014

SIAM J. Control Optim.

Self Cite

View full text Add to dashboard Cite

This paper is addressed to studying the exact controllability for stochastic transport equations by two controls: one is a boundary control imposed on the drift term and the other is an internal control imposed on the diffusion term. By means of the duality argument, this controllability problem can be reduced to an observability problem for backward stochastic transport equations, and the desired observability estimate is obtained by a new global Carleman estimate. Also, we present some results about the lack of exact controllability, which show that the action of two controls is necessary. To some extent, this indicates that the controllability problems for stochastic PDEs differ from their deterministic counterpart.2010 Mathematics Subject Classification. Primary 93B05; Secondary 93B07, 93E20, 60H15.Key Words. Stochastic transport equations, exact controllability, observability, Carleman estimate.Without loss of generality, we assume that 0 ∈ G and 0 =x 1 +x 2 . Put R = max x∈Γ |x| R d . Let

show abstract

Observability estimate and state observation problems for stochastic hyperbolic equations

Cited by 33 publications

References 36 publications

Stochastic Well-Posed Systems and Well-Posedness of Some Stochastic Partial Differential Equations with Boundary Control and Observation

Stochastic Well-Posed Systems and Well-Posedness of Some Stochastic Partial Differential Equations with Boundary Control and Observation

An internal observability estimate for stochastic hyperbolic equations

Exact Controllability for Stochastic Transport Equations

Contact Info

Product

Resources

About