An Approach to the Optimal Time for a Time Optimal Control Problem of an Internally Controlled Heat Equation

Wang, Gengsheng; Zheng, Guojie

doi:10.1137/100793645

Cited by 25 publications

(17 citation statements)

References 10 publications

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“…• There have been some literatures on the approximations of time optimal control problems for the parabolic equations. 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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“…A minimal time control problem is to ask for a control (taking values from a control constraint set which is, in general, a closed and bounded subset in a control space) which drives the corresponding solution of a controlled system from an initial state to a target set in the shortest time, while a minimal norm control problem is to ask for a control which has the minimal norm among all controls that drive the corresponding solutions of a controlled systems from an initial state to a target set at fixed ending time. Several important issues on minimal time (or minimal norm) control problems are as follows: The Pontryagin maximum principle of minimal time (or minimal norm) controls (see, for instance, [8,19,22,24,46]); The existence of minimal time ( or minimal norm) controls (see, for instance, [3,23,34]); Their connections with controllabilities (see, for instance [4,13,30]); Numerical analyses on minimal time (or minimal norm) controls (see, for instance, [12,14,27,37,45]); And the bang-bang property of minimal time (or minimal norm) controls (see, for instance, [6,18,19,22,25,26,28,31,33,36,40,42,43,44,47,49]).…”

Section: Motivationmentioning

confidence: 99%

“…About studies on minimal time and minimal norm control problems, we would like to mention the literatures [2,3,4,6,7,8,9,10,11,13,14,18,19,21,22,23,24,25,26,27,28,30,31,33,34,36,37,40,41,43,44,45,46,47,48,49,50] and the references therein.…”

Section: More About the Bang-bang Propertiesmentioning

confidence: 99%

Decompositions and bang-bang properties

Wang¹,

Zhang²

2017

Mathematical Control &Amp; Related Fields

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Two kinds of important optimal control problems for linear controlled systems are minimal time control problems and minimal norm control problems. A minimal time control problem is to ask for a control (which takes values in a control constraint set in a control space) driving the corresponding solution of a controlled system from an initial state y 0 to a target set in the shortest time, while a minimal norm control problem is to ask for a control which has the minimal norm among all controls driving the corresponding solutions of a controlled system from an initial state y 0 to a target set at fixed ending time T . In this paper, we focus on the case that target sets are the origin of state spaces, control constraint sets are closed balls B(0, M ) in control spaces (centered at the origin and of radius M > 0) and controls are L ∞ functions. The bang-bang property for a minimal time control problem means that any minimal time control, as a function of time, point-wisely takes its value at the boundary of B(0, M ), while the bang-bang property for a minimal norm control problem means that each minimal norm control, as a function of time, point-wisely takes the minimal norm.This paper studies the bang-bang properties for the above-mentioned two kinds of problems from a new perspective. The motivation of this study is as follows: For a given controlled system, a minimal time control problem depends only on (M, y 0 ), while a minimal norm control problem depends only on (T, y 0 ). When a controlled system reads: y ′ (t) = Ay(t) + Bu(t), with (A, B) in R n×n × (R n×m \ {0}), an interesting phenomenon related to the bang-bang properties is as follows: The product space consisting of all pairs (M, y 0 ) can be divided into two disjoint parts so that when (M, y 0 ) is in one part, the corresponding minimal time control problem

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“…Furthermore, convergence of optimal times and controls for u d = 0 with u 0 , u d ∈ H 1/2−ε (Ω) is shown in [23] for a setting with boundary control. More recently, for distributed control and u 0 ∈ H 1 0 (Ω) the error estimate O(h) has been proved in [35] for the linear heat equation and for a semilinear heat equation in [37]. Both articles use a cellwise linear discretization for the control and the set of admissible controls is defined by Q ad := { q ∈ L ∞ ((0, ∞); L 2 (ω)) : q(t) L 2 ≤ 1 a.e.…”

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confidence: 99%