An internal observability estimate for stochastic hyperbolic equations

Fu, Xiaoyu; Liu, Xu; Lü, Qi; Xu, Zuyan

doi:10.1051/cocv/2016042

Cited by 7 publications

(6 citation statements)

References 17 publications

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“…Remark Notice that if a ( x )≡1, then d ( x )=| x − x 0 | 2 , where x 0 is any given point in

double-struckR ∖ [γ, 1]

([, remark 1.1]). If a ( x )= x s ( s ⩾1), then d ( x )= M x , where

M ⩾ \frac{μ_{0}}{s γ^{s - 1}}

.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…If a ( x )= x s ( s ⩾1), then d ( x )= M x , where

M ⩾ \frac{μ_{0}}{s γ^{s - 1}}

. Moreover, it is easy to check that, if d (·) satisfies the condition

(double-struckH)

, then, for any given p ⩾1 and

q \in double-struckR

, the function

\overset{d}{true} (x) = p d (x) + q

still satisfies condition

(double-struckH)

with μ 0 replaced by p μ 0 ([, remark 1.3]). Therefore, throughout this paper, we may assume that d (·) and μ 0 satisfy

μ_{0} > \frac{9 {(T^{*})}^{2}}{M_{0}} and M_{0} ⩾ \max_{x \in [γ, 1]} d (x) .

…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Remark The method in this paper needs to use classical Carleman estimate for non‐degenerate wave equation, the condition T > T ∗ plays a key role in the proof of classical Carleman estimate ( see [, remark 1.4]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Set

φ (x, t) = d (x) - c_{1} {(t - \frac{T}{2})}^{2}

. We recall a known global Carleman estimate for non‐degenerate wave operators ([, theorem 7.1] or [, theorem 4.1]).…”

Section: Observability Of Degenerate Wave Equationsmentioning

confidence: 99%

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Persistent regional null controllability of some degenerate wave equations

Zhang

Gao

2017

Math Methods in App Sciences

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This paper is addressed to a study of the persistent regional null controllability problems for one‐dimensional linear degenerate wave equations through a distributed controller. Different from non‐degenerate wave equations, the classical null controllability results do not hold for some degenerate wave equations. Thus, persistent regional null controllability is introduced, which means finding a control such that the corresponding state of the degenerate wave equation may vanish in a suitable subset of the space domain in a period of time. In order to solve this problem, we need to establish the regional null controllability for degenerate wave equations. This problem is reduced to a suitable observability problem of a linear degenerate wave equation. The key point is to choose a suitable multiplier in order to establish this observability inequality. Copyright © 2017 John Wiley & Sons, Ltd.

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“…Remark Notice that if a ( x )≡1, then d ( x )=| x − x 0 | 2 , where x 0 is any given point in

double-struckR ∖ [γ, 1]

([, remark 1.1]). If a ( x )= x s ( s ⩾1), then d ( x )= M x , where

M ⩾ \frac{μ_{0}}{s γ^{s - 1}}

.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…If a ( x )= x s ( s ⩾1), then d ( x )= M x , where

M ⩾ \frac{μ_{0}}{s γ^{s - 1}}

. Moreover, it is easy to check that, if d (·) satisfies the condition

(double-struckH)

, then, for any given p ⩾1 and

q \in double-struckR

, the function

\overset{d}{true} (x) = p d (x) + q

still satisfies condition

(double-struckH)

with μ 0 replaced by p μ 0 ([, remark 1.3]). Therefore, throughout this paper, we may assume that d (·) and μ 0 satisfy

μ_{0} > \frac{9 {(T^{*})}^{2}}{M_{0}} and M_{0} ⩾ \max_{x \in [γ, 1]} d (x) .

…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Set

φ (x, t) = d (x) - c_{1} {(t - \frac{T}{2})}^{2}

. We recall a known global Carleman estimate for non‐degenerate wave operators ([, theorem 7.1] or [, theorem 4.1]).…”

Section: Observability Of Degenerate Wave Equationsmentioning

confidence: 99%

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Persistent regional null controllability of some degenerate wave equations

Zhang

Gao

2017

Math Methods in App Sciences

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“…In this paper, we obtain the exact controllability of the system (1. where (z, Z, ẑ, Z) solves (3.1) with τ = T and the final datum (z T , ẑT ). Following [9], we can prove that (9.1) holds. Details are too lengthy to be presented here.…”

Section: Further Comments and Open Problemsmentioning

confidence: 95%

Exact Controllability for a Refined Stochastic Wave Equation

Qi¹,

Zhang²

2019

Preprint

Self Cite

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A widely used stochastic wave equation is the classical wave equation perturbed by a term of Itô's integral. We show that this equation is not exactly controllable even if the controls are effective everywhere in both the drift and the diffusion terms and also on the boundary. In some sense this means that some key feature has been ignored in this model. Then, based on a stochastic Newton's law, we propose a refined stochastic wave equation. By means of a new global Carleman estimate, we establish the exact controllability of our stochastic wave equation with three controls. Moreover, we give a result about the lack of exact controllability, which shows that the action of three controls is necessary. Our analysis indicates that, at least from the point of view of control theory, the new stochastic wave equation introduced in this paper is a more reasonable model than that in the existing literatures.

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