Delay systems and optimal control
type: Item
instance of: scholarly article
title: Delay systems and optimal control
zbMATH Open document ID: 1005.49017
DOI: 10.1007/BF02670961
author: X. Xiang
author: Huawu Kuang
published in: Acta Mathematicae Applicatae Sinica. English Series
publication date: 2003-01-01T00:00:00Z
review text: The authors of this paper generalize the Gronwall lemma to the case with time lags and at the same time use it to study controlled delay systems. It is considered the controlled delay systems associated with \(C_0\)-semigroup and analytic semigroup, \[ \dot x(t)=Ax(t)+f(t,x_t,x(t))+B(t)u(t)\quad (0\leq t\leq T),\qquad x(t)=\varphi (t)\quad (-r\leq t\leq 0). \] Here \(\varphi\in C\), \(x_t\in C([-r,a],X)\), \(A:D(A)\to X\) is the infinitesimal generator of a \(C_0-\)semigroup \(T(t)\) (\(t\geq 0\)) satisfying \(|T(t)|_{L}\leq Me^{\omega t}\) (\(M\geq 1\), \(\omega >0\)), \(f\) is Hölder continuous with respect to \(t\) and locally Lipschitz continuous with respect to other arguments. The main result is that there exist mild solutions and optimal control. It is shown a uniqueness result as well. An example, which illustrates the main idea, is discussed.
Mathematics Subject Classification ID: 49K25
Mathematics Subject Classification ID: 49J25
zbMATH DE Number: 1462491
zbMATH Keywords: Gronwall lemma
zbMATH Keywords: controlled delay systems
zbMATH Keywords: optimal control
reviewed by: Drumi Bainov
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02670961
OpenAlex ID: W2083011420
description: scientific article; zbMATH DE number 1462491