Periodic Feedback Stabilization for Linear Periodic Evolution Equations

Wang, Gengsheng; Xu, Yashan

doi:10.1007/978-3-319-49238-4

Cited by 10 publications

(12 citation statements)

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“…• The similar equivalence results in Theorem 1.1 were obtained in [8] for time-invariant systems. Different kinds of characterizations of the periodic stabilization for time-periodic systems have been studied in [5], [6], [9] and [10]. The characterization (for the system (1.1)), given in Theorem 1.1, seems to be new.…”

Section: Main Resultmentioning

confidence: 99%

Characterization by detectability inequality for periodic stabilization of linear time-periodic evolution systems

2020

Preprint

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Section: Main Resultmentioning

confidence: 99%

Characterization by detectability inequality for periodic stabilization of linear time-periodic evolution systems

2020

Preprint

Self Cite

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“…Stabilization of linear autonomous control systems is classically done in finite dimension by poleshifting or by Riccati theory (see, e.g., [25,29,40,43]). In infinite dimension, pole-shifting may be used for some appropriate classes of systems (see [5,10,11], see also [37, page 711] and [48,Chapter 3]), but such approaches rely on spectral considerations and in practice require the numerical computation of eigenelements, which may be hard in general. Riccati theory has also been much explored in infinite dimension (see, e.g., [12,26,27,49] and provides a powerful way for stabilizing a linear control system.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Thanks to the assumptions (H 1 ), (H 2 ), (H 3 ) and (H 4 ), to stabilize (1) it would suffice to focus on the finite-dimensional instable part E ℓ of the infinite-dimensional system (1), as this was done for instance in [5,10,11] (see also [37, page 711] and [48,Chapter 3]). However, in practice eigenelements are not known in general or may be difficult to compute numerically.…”

Section: General Setting and Assumptionsmentioning

confidence: 99%

Stabilization of infinite-dimensional linear control systems by POD reduced-order Riccati feedback

Trélat¹,

Wang²,

Xu³

2019

Preprint

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There exist many ways to stabilize an infinite-dimensional linear autonomous control systems when it is possible. Anyway, finding an exponentially stabilizing feedback control that is as simple as possible may be a challenge. The Riccati theory provides a nice feedback control but may be computationally demanding when considering a discretization scheme. Proper Orthogonal Decomposition (POD) offers a popular way to reduce large-dimensional systems. In the present paper, we establish that, under appropriate spectral assumptions, an exponentially stabilizing feedback Riccati control designed from a POD finite-dimensional approximation of the system stabilizes as well the infinite-dimensional control system.

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“…Some necessary and sufficient conditions for the stabilization of a class of linear distributed system were studied in these two works. It deserves to mention what follows: Since time-periodic feedback laws are much more powerful than time-invariant feedback laws (see [35]), the authors of [32,19,27] could expect the feedback laws stabilizing systems with bigger sampling periods. As payments, the cost of using such feedback laws is higher than that of using time-invariant feedback laws.…”

Section: Aims Previous Work and Motivationsmentioning

confidence: 99%

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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Periodic Feedback Stabilization for Linear Periodic Evolution Equations

Cited by 10 publications

References 0 publications

Characterization by detectability inequality for periodic stabilization of linear time-periodic evolution systems

Characterization by detectability inequality for periodic stabilization of linear time-periodic evolution systems

Stabilization of infinite-dimensional linear control systems by POD reduced-order Riccati feedback

Output feedback stabilization for heat equations with sampled-data controls

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