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Regularity of Multipliers for Multiobjective Optimal Control Problems Governed by Evolution Equations

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Abstract

This article is concerned with multiobjective optimal control problems, driven by evolution equations, and involving implicit control constraints and mixed pointwise control-state constraints in infinite dimensional separable Banach spaces. We consider bounded controls and inclusion-type mixed pointwise constraints, which are given in terms of measurable set-valued mappings whose images are closed convex sets with nonempty interior. In this case, the multiplier associated with mixed constraints is an element of the dual space of the Banach space-valued essentially bounded functions. Exploiting the combination of the Ioffe-Levin decomposition theorem for this dual and a Lagrange multiplier theorem obtained for an abstract multi-criteria optimization framework, we set up Fritz-John necessary optimality conditions, in the presence of integrable functions and singular measures as multipliers, for local weak Pareto solutions of the problems under investigation. Moreover, we give some conditions under which the multipliers are regular. An example of application of the main result is also provided to show the existence of a nontrivial singular measure.

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Acknowledgements

This research is funded by University of Economics Ho Chi Minh City, Vietnam. The author would like to thank the editor and the referee for their careful reading, valuable remarks and suggestions, which have helped him to improve the paper.

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  1. Department of Mathematics and Statistics, and Institute of Applied Mathematics, University of Economics Ho Chi Minh City, 59C Nguyen Dinh Chieu, D.3, Ho Chi Minh City, Vietnam

    Tuan Nguyen Dinh

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Communicated by Nguyen Dong Yen.

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Nguyen Dinh, T. Regularity of Multipliers for Multiobjective Optimal Control Problems Governed by Evolution Equations. J Optim Theory Appl 196, 762–796 (2023). https://doi.org/10.1007/s10957-022-02143-7

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