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Minimal Time Impulse Control of an Evolution Equation

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Yueliang Duan
Yueliang Duan
Yueliang Duan
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Lijuan Wang
Lijuan Wang
Lijuan Wang
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Can Zhang at Wuhan University
Can Zhang
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Abstract

This paper is concerned with a kind of minimal time control problem for a linear evolution equation with impulse controls. Each problem depends on two parameters: the upper bound of the control constraint and the moment of impulse time. The purpose of such a problem is to find an optimal impulse control (among certain control constraint set), which steers the solution of the evolution equation from a given initial state to a given target set as soon as possible. In this paper, we study the existence of optimal control for this problem; by the geometric version of the Hahn–Banach theorem, we show the bang–bang property of optimal control, which leads to the uniqueness of the optimal control; we also establish the continuity of the minimal time function of this problem with respect to the above mentioned two parameters, and discuss the convergence of the optimal control when the two parameters converge.
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Journal of Optimization Theory and Applications (2019) 183:902–919
https://doi.org/10.1007/s10957-019-01552-5
Minimal Time Impulse Control of an Evolution Equation
Yueliang Duan1·Lijuan Wang2·Can Zhang1
Received: 19 December 2018 / Accepted: 8 June 2019 / Published online: 8 July 2019
© Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract
This paper is concerned with a kind of minimal time control problem for a linear evo-
lution equation with impulse controls. Each problem depends on two parameters: the
upper bound of the control constraint and the moment of impulse time. The purpose of
such a problem is to find an optimal impulse control (among certain control constraint
set), which steers the solution of the evolution equation from a given initial state to a
given target set as soon as possible. In this paper, we study the existence of optimal
control for this problem; by the geometric version of the Hahn–Banach theorem, we
show the bang–bang property of optimal control, which leads to the uniqueness of
the optimal control; we also establish the continuity of the minimal time function of
this problem with respect to the above mentioned two parameters, and discuss the
convergence of the optimal control when the two parameters converge.
Keywords Minimal time control ·Impulse control ·Bang–bang property
Mathematics Subject Classification 49J15 ·49J20 ·49K15 ·49K20
Communicated by Roland Herzog.
BCan Zhang
canzhang@whu.edu.cn
Yueliang Duan
duanyl@csu.edu.cn
Lijuan Wang
ljwang.math@whu.edu.cn
1School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
2School of Mathematics and Statistics, Computational Science Hubei Key Laboratory,
Wuhan University, Wuhan 430072, China
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... In [10], Kunisch and Rao investigated the Hamilton-Jacobi-Bellman approach for a minimal time problem with impulse controls governed by ordinary differential equations. In [4] and [24], Duan and Wang considered optimal time impulse control problems of the linear and semilinear heat equations, and proved that the optimal controls exhibit the bang-bang property. ...
... Pontryagin's Maximum Principle. Similar to Theorem 2.1 in [4], we know that there exists at least one optimal control f * = {f j } ∞ j=1 ∈ F for (T P). In the following, we will present necessary and sufficient conditions of the optimal control for (T P). ...
... We first show the proof of (1). The proof of the necessity is similar to Theorem 4.1 of Chapter 7 in [13] or Theorem 2.1 in [4], and we omit its proof. For the bang-bang and unique properties, we refer the readers to Theorem 2.1 in [4]. ...
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... For time-dependent systems, the control problem (1.1)-(1.2) posed in M(Ī c ; L 2 (ω)) yields controls with compact support in time. This characteristic allows for determining the optimal moments for control device actions, akin to a generalization of impulse control [31][32][33][34][35][36][37][38][39][40][41]. Recall that in impulse control problems, the control q in (1.1)-(1.2) is replaced by ...
... [9]). This is exactly the impulse control problem, studied extensively in the literature, e.g., [33][34][35][36][37], and the references therein. ...
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... Introduction. Recently, control theory has found extensive applications in various fields including engineering, biology, medicine, physics, economics, computers and ecology, especially in systems with impulse control, feedback control and optimal control (see [1,9,12,21,37,38,40,41,42,43]). ...
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... A minimal norm impulse control problem for parabolic equation consists in finding an impulse control, which has the minimal norm among all controls steering solutions (of the controlled equation) from a given initial state to a given target set in a fixed time interval. To the best of our knowledge, a minimal time impulse control problem for a linear evolution was studied in [13], where the existence and the bang-bang property of optimal control, the continuity of the minimal time function (with respect to two parameters: the upper bound of the control constraint and the moment of impulse time) and the convergence of optimal control (when the above-mentioned parameters converge) were discussed; a minimal norm impulse control problem for a semilinear heat equation was studied in [12], where the existence of optimal controls and a nontrivial Pontryagin's maximum principle were established. In this paper, we will consider a kind of minimal time impulse control problem for a semilinear heat equation. ...
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... Among many existing literature on norm optimal control problems for parabolic equations, the control inputs usually affect the control systems at each instant of time (see, for instance, [1][2][3][4]). Impulse control is a kind of important control and has wide applications (see, for instance, [5][6][7][8]). In many cases, an impulse control can give an efficient way to deal with systems, which cannot endure continuous control inputs. ...
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