Null controllability of some degenerate wave equations

Zhang, Muming; Gao, Hang

doi:10.1007/s11424-016-5281-3

Cited by 25 publications

(16 citation statements)

References 15 publications

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“…In the neighborhood of an endpoint x =0 of this string, the elastic is sufficiently small or the linear density is large enough. Moreover, by [, theorem 2.1], for any

(y_{0}, y_{1}) \in H_{a}^{1} (normal Ω) \times L^{2} (normal Ω)

and

h \in L^{2} (ω \times (0, T^{'}))

, admits a unique weak solution

y \in C ([0, T^{'}]; H_{a}^{1} (normal Ω)) \cap C^{1} ([0, T^{'}]; L^{2} (normal Ω))

.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Further, if system is null controllable in time T , and we stop controlling the system at time T , then for all t ⩾ T , it holds that y ( t )= y t ( t )=0 in Ω. However, the null controllability does not hold for some degenerate wave equations (e.g., a ( x )= x s , s ⩾2; [, theorem 1.2] or [, example 3.7,3.8]), that is, we cannot find any control such that the corresponding state of the degenerate wave equation equal to zero in the whole of space domain. Therefore, persistent regional null controllability is introduced.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…However, very little results are known about the controllability of degenerate wave equations. We refer to . The persistent regional null controllability of degenerate heat equation was discussed in .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Proof By [, theorem 1.2], for any

(v_{T}^{0}, v_{T}^{1}) \in L^{2} (normal Ω) \times H_{a}^{*} (normal Ω)

, the system admits a unique solution

v \in C (0, T; L^{2} (normal Ω)) \cap C^{1} (0, T; H_{a}^{*} (normal Ω))

in the sense of transposition.…”

Section: Regional Null Controllability Problem and Its Transformationmentioning

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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“…In the neighborhood of an endpoint x =0 of this string, the elastic is sufficiently small or the linear density is large enough. Moreover, by [, theorem 2.1], for any

(y_{0}, y_{1}) \in H_{a}^{1} (normal Ω) \times L^{2} (normal Ω)

and

h \in L^{2} (ω \times (0, T^{'}))

, admits a unique weak solution

y \in C ([0, T^{'}]; H_{a}^{1} (normal Ω)) \cap C^{1} ([0, T^{'}]; L^{2} (normal Ω))

.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Proof By [, theorem 1.2], for any

(v_{T}^{0}, v_{T}^{1}) \in L^{2} (normal Ω) \times H_{a}^{*} (normal Ω)

, the system admits a unique solution

v \in C (0, T; L^{2} (normal Ω)) \cap C^{1} (0, T; H_{a}^{*} (normal Ω))

in the sense of transposition.…”

Section: Regional Null Controllability Problem and Its Transformationmentioning

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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“…Although the controllability of infinite dimensional systems has been studied extensively, e.g., null controllability [9,10], local controllability to trajectories [11][12][13][14], approximate controllability [15,16], exact controllability [17,18], and boundary controllability [19], control of the Burgers-Fisher equation is still in its infancy and remains open. In this paper, we will deal with the local exact controllability to the trajectories of the Burgers-Fisher equation.…”

Section: Introductionmentioning

confidence: 99%

Local Exact Controllability to the Trajectories of Burgers–Fisher Equation

Shi

2020

Mathematical Problems in Engineering

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This paper is addressed to study local exact controllability to the trajectories of the Burgers–Fisher (BF) equation. By using the global Carleman estimate for the second-order parabolic operator, we establish the observable inequality and obtain the exact controllability to the trajectories of the linear system. Then, by local inverse theory, we consider the controllability result for the Burgers–Fisher equation.

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The energy decay rate of a transmission system governed by the degenerate wave equation with drift and under heat conduction with the memory effect

Akil,

Fragnelli,

Issa

2024

Mathematische Nachrichten

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In this paper, we investigate the stabilization of the transmission problem of the degenerate wave equation and the heat equation under the Coleman–Gurtin heat conduction law or Gurtin–Pipkin law with the memory effect. We investigate the polynomial stability of this system when employing the Coleman–Gurtin heat conduction, establishing a decay rate of type . Next, we demonstrate exponential stability in the case when Gurtin–Pipkin heat conduction is applied.

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Null controllability of some degenerate wave equations

Cited by 25 publications

References 15 publications

Persistent regional null controllability of some degenerate wave equations

Persistent regional null controllability of some degenerate wave equations

Local Exact Controllability to the Trajectories of Burgers–Fisher Equation

The energy decay rate of a transmission system governed by the degenerate wave equation with drift and under heat conduction with the memory effect

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