Impulse output rapid stabilization for heat equations

Phung, Kim Dang; Wang, Gengsheng; Xu, Yashan

doi:10.1016/j.jde.2017.06.008

Cited by 36 publications

(35 citation statements)

References 26 publications

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“…By the same arguments to those used to prove properties of the problem (SNP ) in the last subsection, we can get the results in the next theorem. (These results are quite similar to those in Theorem 3.4 in [25].)…”

Section: Minimal Norm Impulse Control Problemssupporting

confidence: 89%

“…Unfortunately, there has been no systematic theory on this subject. Most of publications on this issue focus on how to construct output feedback laws to stabilize some special equations (see [13,6,22,1,25] and references therein). The current paper also only provides a way to design an output feedback law.…”

Section: Aims Previous Work and Motivationsmentioning

confidence: 99%

“…Define a time-invariant control: 14), we can use a standard way (see the proof of Theorem 3.1 in [25]) to show thatũ satisfies (2.2) with the above-mentioned C(ε, M, T 2 − T 1 ). So the equation (2.1) has the partially null approximate controllability with a cost.…”

Section: )mentioning

confidence: 99%

“…It was proven that the equation (3.18) has the null approximate controllability with a cost (see Theorem 3.1 in [25]). Consequently, it has the partially null approximate controllability with a cost.…”

Section: Minimal Norm Impulse Control Problemsmentioning

confidence: 99%

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Output feedback stabilization for heat equations with sampled-data controls

Lin

Liu

Wang

2020

Journal of Differential Equations

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In this paper, we build up an output feedback law to stabilize a sampled-data controlled heat equation (with a potential) in a bounded domain Ω. The feedback law abides the following rules: First, we divide equally the time interval [0, +∞) into infinitely many disjoint time periods, and divide each time period into three disjoint subintervals. Second, for each time period, we observe a solution over an open subset of Ω in the first subinterval; take sample from outputs at one time point of the first subinterval; add a time-invariant output feedback control over another open subset of Ω in the second subinterval; let the equation evolve free in the last subinterval. Thus, the corresponding feedback control is of sampleddata. Our feedback law has the following advantages: the sampling period (which is the length of the above time period) can be arbitrarily taken; the feedback law has an explicit expression in terms of the sampling period; the behaviors of the norm of the feedback law, when the sampling period goes to zero or infinity, are clear. The construction of the feedback law is based on two kinds of approximate null-controllability for heat equations. One has time-invariant controls, while another has impulse controls. The studies of the aforementioned controllability with time-invariant controls need a new observability inequality for heat equations built up in the current work.

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Section: Minimal Norm Impulse Control Problemssupporting

confidence: 89%

Section: Aims Previous Work and Motivationsmentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: Minimal Norm Impulse Control Problemsmentioning

confidence: 99%

See 2 more Smart Citations

Output feedback stabilization for heat equations with sampled-data controls

Lin

Liu

Wang

2020

Journal of Differential Equations

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“…The function V (y) = y, C −1 λ y is a Lyapunov function (as noticed in [5]), and the feedback K λ yields exponential stability with rate −λ. We also refer to [6,25] for developments on rapid stabilization.…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

Characterization by observability inequalities of controllability and stabilization properties

Trélat

Wang

2020

Pure Appl. Analysis

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Given a linear control system in a Hilbert space with a bounded control operator, we establish a characterization of exponential stabilizability in terms of an observability inequality. Such dual characterizations are well known for exact (null) controllability. Our approach exploits classical Fenchel duality arguments and, in turn, leads to characterizations in terms of observability inequalities of approximately null controllability and of α-null controllability. We comment on the relationships between those various concepts, at the light of the observability inequalities that characterize them.

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