Stability and solvability for a class of optimal control problems described by non-instantaneous impulsive differential
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(2020) 2020:524
RESEARCH
Open Access
Stability and solvability for a class of optimal control problems described by non-instantaneous impulsive differential equations Yi Chen1*
and Kaixuan Meng1
*
Correspondence: [email protected]; [email protected] 1 School of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, P.R. China
Abstract In this paper, we investigate the existence and stability of solutions for a class of optimal control problems with 1-mean equicontinuous controls, and the corresponding state equation is described by non-instantaneous impulsive differential equations. The existence theorem is obtained by the method of minimizing sequence, and the stability results are established by using the related conclusions of set-valued mappings in a suitable metric space. An example with the measurable admissible control set, in which the controls are not continuous, is given in the end. MSC: 34H05; 49J15; 34D20 Keywords: Optimal control; Non-instantaneous impulse; Stability analysis
1 Introduction Impulsive phenomena are results of a sudden change in the state of system due to external interference; they often occur in nature and human activities. According to the duration of the change, the phenomena of this kind are divided into two categories. One is that the duration of this change is relatively short compared with the total duration of the whole process, which is called instantaneous impulse. The other is that they start from any fixed point and remain active for a limited time interval, namely the effects are continuous. We call it non-instantaneous impulse (see [1–8]). Most of the mathematical models extracted from impulsive phenomena are characterized by impulsive differential equations, which can be classified into two categories in accordance with the type of impulse: instantaneous impulsive differential equations and non-instantaneous ones. Now, a large number of references deal with the impulsive differential equations. By the type of impulse, they include the non-instantaneous case [1–12] and the instantaneous case [13–22]. This paper is devoted to the study of the differential equation with impulse of noninstantaneous type on account of its reality and significance. For instance, the state change © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy
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