Attainable Subspaces and the Bang-Bang Property of Time Optimal Controls for Heat Equations

Wang, Gengsheng; Xu, Yashan; Zhang, Yubiao

doi:10.1137/140966022

Cited by 29 publications

(24 citation statements)

References 38 publications

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“…We refer to [15,41] for semi-discrete finite element approximations, and [34,43] for perturbations of equations. About more works on time optimal control problems, we would like to mention [2,10,11,16,17,18,21,22,25,27,30,31,35,37,38,39,40,42,44] and the references therein.…”

Section: Resultsmentioning

confidence: 99%

Time optimal sampled-data controls for the heat equation

Wang

Yang

Zhang

2017

Comptes Rendus. Mathématique

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In this paper, we first design a time optimal control problem for the heat equation with sampled-data controls, and then use it to approximate a time optimal control problem for the heat equation with distributed controls. Our design is reasonable from perspective of sampled-data controls. And it might provide a right way for the numerical approach of a time optimal distributed control problem, via the corresponding semi-discretized (in time variable) time optimal control problem.The study of such a time optimal sampled-data control problem is not easy, because it may have infinitely many optimal controls. We find connections among this problem, a minimal norm sampled-data control problem and a minimization problem. And obtain some properties on these problems. Based on these, we not only build up error estimates for optimal time and optimal controls between the time optimal sampled-data control problem and the time optimal distributed control problem, in terms of the sampling period, but also prove that such estimates are optimal in some sense.

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Section: Resultsmentioning

confidence: 99%

Time optimal sampled-data controls for the heat equation

Wang

Yang

Zhang

2017

Comptes Rendus. Mathématique

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“…The proof of Lemma 4.4 is similar to that of Lemma 2.1 in [21]. For the sake of completeness, we give its detailed proof below.…”

Section: Properties Of Functionalsmentioning

confidence: 89%

Optimal control problem for exact synchronization of parabolic system

Wang¹,

Yan²

2019

Mathematical Control &Amp; Related Fields

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This paper studies a kind of minimal time control problems related to the exact synchronization for a controlled linear system of parabolic equations. Each problem depends on two parameters: the bound of controls and the initial state. The purpose of such a problem is to find a control (from a constraint set) synchronizing components of the corresponding solution vector for the controlled system in the shortest time. In this paper, we build up a necessary and sufficient condition for the optimal time and the optimal control; we also obtain how the existence of optimal controls depends on the above mentioned two parameters.2010 Mathematics Subject Classifications. 49K20, 93B05, 93B07, 93C20 y(t; y 0 , u) = (y 1 (t; y 0 , u), y 2 (t; y 0 , u), . . . , y n (t; y 0 , u)) ⊤ for the solution of the system (1.1). (Here and throughout this paper, we denote the transposition of a matrix J by J ⊤ .) It is well known that for each T > 0, y(·; y 0 , u) ∈ W 1,2 (0, T ; H −1 (Ω) n ) ∩ L 2 (0, T ; H 1 0 (Ω) n ) ⊆ C([0, T ]; L 2 (Ω) n ). We will treat this solution as a function from [0, +∞) to L 2 (Ω) n .We next define control constraint set U M (with M > 0) and the target set S as follows:S {(y 1 , y 2 , . . . , y n ) ⊤ ∈ L 2 (Ω) n : y 1 = y 2 = · · · = y n }.Given M > 0, y 0 ∈ L 2 (Ω) n , we define the minimal time control problem (T P ) y 0 M :T (M, y 0 ) inf u∈U M {T ≥ 0 : u(·) = 0 and y(·; y 0 , u) ∈ S over [T, +∞)}.About Problem (T P ) y 0 M , several notes are given in order: (a 1 ) We call T (M, y 0 ) the optimal time; we call u ∈ U M an admissible control if there is T ≥ 0 so that u(·) = 0 and y(·; y 0 , u) ∈ S over [T, +∞); we call u * ∈ U M an optimal control if T (M, y 0 ) < +∞, u * (·) = 0 and y(·; y 0 , u * ) ∈ S over [T (M, y 0 ), +∞); we agree that T (M, y 0 ) = +∞ if the problem (T P ) y 0 M has no admissible control.(a 2 ) One can easily check that if y = (y 1 , y 2 , . . . , y n ) ⊤ ∈ L 2 (Ω) n , then y ∈ S if and only if Dy = 0, where and throughout this paper,

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“…The bang-bang property has been studied in much work (see, e.g., [1,11,13,14,[29][30][31][32]). However, regarding the time-varying bang-bang property, particularly in infinitedimensional cases, there is only one paper [19] in which the authors considered a kind of time optimal control problem that involves an interior subset.…”

Section: B(0 R))mentioning

confidence: 99%

On a kind of time optimal control problem of the heat equation

Zhang

2018

Adv Differ Equ

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In this paper, we consider a kind of time-varying bang-bang property of time optimal boundary controls for the heat equation. The time-varying bang-bang property in the interior domain has been considered in some papers, but regarding the time optimal boundary control problem it is still unsolved. In this paper, we determine that there exists at least one solution to the time optimal boundary control problem with time-varying controls. MSC: 35K05; 49J20

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Attainable Subspaces and the Bang-Bang Property of Time Optimal Controls for Heat Equations

Cited by 29 publications

References 38 publications

Time optimal sampled-data controls for the heat equation

Time optimal sampled-data controls for the heat equation

Optimal control problem for exact synchronization of parabolic system

On a kind of time optimal control problem of the heat equation

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