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Controlled singular evolution equations and Pontryagin type maximum principle with applications

This work was supported by a key project of the International Science and Technology Cooperation Program of Shaanxi Research and Development Plan (2019KWZ-08). The work of J.J. Nieto has been partially supported by the Agencia Estatal de Investigacion (AEI) of Spain under Grant MTM2016-75140-P and co-financed by European Community fund FEDER and by Xunta de Galicia, grant ED431C 2019/02. This research was partially supported by the Portuguese Foundation for Science and Technology (FCT) within Project n. 147 Controlo Otimo e Modelacao Matematica da Pandemia COVID-19: contributos para uma estrategia sistemica de intervencao em saude na omunidade, in the scope of the RESEARCH 4 COVID-19 call financed by FCT; and by the Instituto de Salud Carlos III, within the Project COV20-00617 Prediccion dinamica de escenarios de afectacion por COVID-19 a corto y medio plazo (PREDICO), in the scope of the Fondo COVID financed by the Ministerio de Ciencia e Innovacion of Spain

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  • 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.

    Mathematics Subject Classification: Primary: 49J20; Secondary: 34A08.


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