Painlevé-Kuratowski convergences of solutions to nonlinear multiobjective optimal control problems

Lam Quoc Anh; Vo Thanh Tai; Tran Ngoc Tam

doi:10.3934/eect.2024024

Article Contents

2024, Volume 13, Issue 4: 1229-1249. Doi: 10.3934/eect.2024024

This issue Previous Article Approximation of the set of integrable trajectories of the control system with $ L_2 $ norm constraints on control functions Next Article

Painlevé-Kuratowski convergences of solutions to nonlinear multiobjective optimal control problems

1.
Department of Mathematics, Teacher College, Can Tho University, Viet Nam
2.
Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Viet Nam
3.
Department of Mathematics, Faculty of Education, An Giang University, Viet Nam
4.
Vietnam National University, Ho Chi Minh City, Viet Nam
5.
Department of Mathematics, College of Natural Sciences, Can Tho University, Viet Nam

^*Corresponding author: Vo Thanh Tai
^*Corresponding author: Vo Thanh Tai

Received: December 2023

Early access: April 29, 2024

Published online: April 29, 2024

This research is funded by Vietnam National University HoChiMinh City (VNU-HCM) under grant number C2022-16-05.

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Abstract

In this paper, we consider nonlinear multiobjective optimal control problems and investigate the convergence conditions of their feasible solution sets and efficient solution sets. Firstly, we utilize the Lipschitz continuity of the function on the right-hand side of the state equation and the uniform continuity of the objective map to analyze the convergence of the sequence of feasible solution sets in the sense of Painlevé-Kuratowski and the continuous convergence of the sequence of objective maps. Subsequently, we employ these obtained results in conjunction with the properties of the generalized convexity integrand of the objective map to formulate the upper and lower Painlevé-Kuratowski convergence of efficient solution sets. Furthermore, when the data of the reference problems do not satisfy the generalized convexity conditions, we employ the induction condition along with a key hypothesis to address the upper and lower Painlevé-Kuratowski convergence of the efficient solution sets. Finally, as applications of the obtained results, we discuss convergence conditions for two practical scenarios involving the glucose and epidemic models.

Keywords:

Multiobjective optimal control problems,
Painlevé-Kuratowski convergence,
convexity integrand,
equimeasurability,
glucose/epidemic model.

Mathematics Subject Classification: 49K40, 90B50, 93C15.

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