ArticlePublisher preview available

Minimal Time Impulse Control Problem of Semilinear Heat Equation

Authors:
Lijuan Wang
Lijuan Wang
Lijuan Wang
  • This person is not on ResearchGate, or hasn't claimed this research yet.
Read publisher preview
Download citation
To read the full-text of this research, you can request a copy directly from the author.

Abstract

The paper is concerned with a kind of minimal time impulse control problem for a semilinear heat equation. We study the existence of optimal controls of this problem, establish a nontrivial Pontryagin’s maximum principle for this problem and then derive the bang–bang property of optimal controls. Based on the existence and the bang–bang property of optimal controls, we discuss the equivalence of the minimal time impulse control problem and its corresponding minimal norm impulse control problem.
A preview of this full-text is provided by Springer Nature.
Learn more
Content available from Journal of Optimization Theory and Applications
This content is subject to copyright. Terms and conditions apply.
Journal of Optimization Theory and Applications (2021) 188:805–822
https://doi.org/10.1007/s10957-020-01807-6
Minimal Time Impulse Control Problem of Semilinear Heat
Equation
Lijuan Wang1
Received: 7 September 2020 / Accepted: 31 December 2020 / Published online: 27 January 2021
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature 2021
Abstract
The paper is concerned with a kind of minimal time impulse control problem for a
semilinear heat equation. We study the existence of optimal controls of this problem,
establish a nontrivial Pontryagin’s maximum principle for this problem and then derive
the bang–bang property of optimal controls. Based on the existence and the bang–bang
property of optimal controls, we discuss the equivalence of the minimal time impulse
control problem and its corresponding minimal norm impulse control problem.
Keywords Minimal time control ·Minimal norm control ·Impulse control ·
Pontryagin’s maximum principle
Mathematics Subject Classification 49K20 ·49J20 ·93C20
1 Introduction
Among many existing literature on time and norm optimal control problems for
parabolic equations, the control inputs usually affect the control systems at each instant
of time (see, for instance, [1–11]). Impulse control is a kind of important control and
has wide applications (see, for instance, [12–16]). In many cases, an impulse control
can give an efficient way to deal with systems, which cannot endure continuous con-
trol inputs. For example, relevant control for acting on a population of bacteria may
be impulsive so that the density of the bacteria may change instantaneously. Other-
wise, continuous control would enhance drug resistance of bacteria (see, for instance,
[14,15]).
Communicated by Michael Hinze.
BLijuan Wang
ljwang.math@whu.edu.cn
1School 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 fact, the theory of minimum time impulse controls has been extensively used in various practical mathematical models, and has become an important research field in recent years. This can be found in a large number of literature (see [1,4,7,11,12,15,16,21,23,26,28,31]). ...
Minimal Time Impulse Control for a Class of Homogeneous Evolution Equations
Article
Full-text available
  • Jul 2024
  • J DYN CONTROL SYST
In this paper, we deal with a minimal time control problem for a class of homogeneous evolution equations with impulse controls. We provide a systematical method to study the existence, uniqueness and bang-bang property of optimal controls for the problem. Moreover, we also establish the continuous dependence of the optimal control by the continuity of the minimal time function of this problem with respect to the parameters.
View
... 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. ...
View
View
Minimal Norm Control Problem Governed by Semilinear Heat Equation with Impulse Control
Article
Full-text available
  • Feb 2020
  • J OPTIMIZ THEORY APP
This paper studies a kind of minimal norm optimal control problem for a semilinear heat equation with impulse controls; it consists in finding an impulse control, which has the minimal norm among all controls steering solutions of the controlled system to a given target from an initial state in a fixed time interval. We will study the existence of optimal controls to this problem. Then, we will establish a nontrivial Pontryagin’s maximum principle of this problem.
View
Minimal Time Impulse Control of an Evolution Equation
Article
Full-text available
  • Dec 2019
  • J OPTIMIZ THEORY APP
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.
View
Computation of Time Optimal Control Problems Governed by Linear Ordinary Differential Equations
Article
Full-text available
  • Oct 2017
  • J SCI COMPUT
In this paper, a novel numerical algorithm is presented to compute the optimal time of a time optimal control problem where the governing system is a linear ordinary differential equation. By the equivalence between time optimal control problem and norm optimal control problem, computation of the optimal time can be obtained by solving a sequence of norm optimal control problems, which are transferred into their Lagrangian dual problems. The nonsmooth structure of the dual problem is approximated by the iteratively reweighted least square strategy. Several numerical tests are given to show the efficiency of the proposed algorithm.
View
Equivalence between Minimal Time and Minimal Norm Control Problems for the Heat Equation
Article
Full-text available
  • Sep 2016
This paper presents the equivalence between minimal time and minimal norm control problems for internally controlled heat equations. The target is an arbitrarily fixed bounded, closed and convex set with a nonempty interior in the state space. This study differs from [G. Wang and E. Zuazua, \textit{On the equivalence of minimal time and minimal norm controls for internally controlled heat equations}, SIAM J. Control Optim., 50 (2012), pp. 2938-2958] where the target set is the origin in the state space. When the target set is the origin or a ball, centered at the origin, the minimal norm and the minimal time functions are continuous and strictly decreasing, and they are inverses of each other. However, when the target is located in other place of the state space, the minimal norm function may be no longer monotonous and the range of the minimal time function may not be connected. These cause the main difficulty in our study. We overcome this difficulty by borrowing some idea from the classical raising sun lemma (see, for instance, Lemma 3.5 and Figure 5 on Pages 121-122 in [E. M. Stein and R. Shakarchi, \textit{Real Analysis: Measure Theory, Integration, and Hilbert Spaces}, Princeton University Press, 2005]).
View
Impulse and Sampled-Data Optimal Control of Heat Equations, and Error Estimates
Article
Full-text available
  • Sep 2015
We consider the optimal control problem of minimizing some quadratic functional over all possible solutions of an internally controlled multi-dimensional heat equation with a periodic terminal state constraint. This problem has a unique optimal solution, which can be characterized by an optimality system derived from the Pontryagin maximum principle. We define two approximations of this optimal control problem. The first one is an impulse approximation, and consists of considering a system of linear heat equations with impulse control. The second one is obtained by the sample-and-hold procedure applied to the control, resulting into a sampled-data approximation of the controlled heat equation. We prove that both problems have a unique optimal solution, and we establish precise error estimates for the optimal controls and optimal states of the initial problem with respect to its impulse and sampled-data approximations.
View
View
Minimal time control of exact synchronization for parabolic systems
Article
  • Mar 2018
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.
View
On the bang-bang control problem
Article
  • Mar 1955
Let S be a physical system whose state at any time is described by an n-dimensional vector x(t), where x(t) is determined by a linear differential equation Z = Az, with A a constant matrix. Application of external influences will yield an inhomogeneous equation, Z = Az + f, where f, the 'forcing term', represents the control. A problem of some importance in the theory of control circuits is that of choosing f so as to reduce z to 0 in minimum time. If f is restricted to belong to the class of vectors whose i(th) components can assume only the values =b sub i, the control is said to be of the 'bang-bang' type. The case where all the solutions of Z = Az approach zero as t approaches infinity is considered.
View
Lecture Notes in Control and Information Sciences
Book
  • Jan 2012
This book is devoted to analysis and design on delta operator systems. When sampling is fast, a dynamical system will become difficult to control, which can be seen in wide real world applications. Delta operator approach is very effective to deal with fast sampling systems. Moreover, it is easy to observe and analyze the control effect with different sampling periods in delta operator systems. The framework of this book has been carefully constructed for delta operator systems to handle sliding mode control, time delays, filter design, finite frequency and networked control. These problems indeed are especially important and significant in automation and control systems design. Through the clear framework of the book, readers can easily go through the learning process on delta operator systems via a precise and comfortable learning sequence. Following this enjoyable trail, readers will come out knowing how to use delta operator approach to deal with control problems under fast sampling case. This book should be a good reference for academies, post-graduates scientists and engineers working in the field of control science and control engineering.
View
Bang-bang property of time optimal controls for some semilinear heat equation
Article
  • Oct 2015
In this paper, we derive a bang-bang property of a kind of time optimal control problem for some semilinear heat equation on bounded C2C^2 domains (of the Euclidean space), with homogeneous Dirichlet boundary condition and controls distributed on an open and non-empty subset of the domain where the equation evolves.
View