Open AccessControlled singular evolution equations and Pontryagin type maximum principle with applications
Author(s) -
Xiaoli Ding,
Iván Area,
Juan J. Nieto
Publication year - 2022
Publication title -
evolution equations and control theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.665
H-Index - 19
eISSN - 2163-2480
pISSN - 2163-2472
DOI - 10.3934/eect.2021059
Subject(s) - pontryagin's minimum principle , maximum principle , optimal control , mathematics , type (biology) , singular control , mathematical optimization , ecology , biology
Due to the propagation of new coronavirus (COVID-19) on the community, global researchers are concerned with how to minimize the impact of COVID-19 on the world. Mathematical models are effective tools that help to prevent and control this disease. This paper mainly focuses on the optimal control problems of an epidemic system governed by a class of singular evolution equations. The mild solutions of such equations of Riemann-Liouville or Caputo types are special cases of the proposed equations. We firstly discuss well-posedness in an appropriate functional space for such equations. In order to reduce the cost caused by control process and vaccines, and minimize the total number of susceptible people and infected people as much as possible, an optimal control problem of an epidemic system is presented. And then for associated control problem, we use a generalized Liapunov type theorem and the spike perturbation technique to obtain a Pontryagin type maximum principle for its optimal controls. In order to derive the maximum principle for an optimal control problems, some techniques from analytical semigroups are employed to overcome the difficulties. Finally, we discuss the potential applications.