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Optimization of Mayer Problem with Sturm–Liouville-Type Differential Inclusions

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Abstract

The present paper studies a new class of problems of optimal control theory with Sturm–Liouville-type differential inclusions involving second-order linear self-adjoint differential operators. Our main goal is to derive the optimality conditions of Mayer problem for differential inclusions with initial point constraints. By using the discretization method guaranteeing transition to continuous problem, the discrete and discrete-approximation inclusions are investigated. Necessary and sufficient conditions, containing both the Euler–Lagrange and Hamiltonian-type inclusions and “transversality” conditions are derived. The idea for obtaining optimality conditions of Mayer problem is based on applying locally adjoint mappings. This approach provides several important equivalence results concerning locally adjoint mappings to Sturm–Liouville-type set-valued mappings. The result strengthens and generalizes to the problem with a second-order non-self-adjoint differential operator; a suitable choice of coefficients then transforms this operator to the desired Sturm–Liouville-type problem. In particular, if a positive-valued, scalar function specific to Sturm–Liouville differential inclusions is identically equal to one, we have immediately the optimality conditions for the second-order discrete and differential inclusions. Furthermore, practical applications of these results are demonstrated by optimization of some “linear” optimal control problems for which the Weierstrass–Pontryagin maximum condition is obtained.

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Acknowledgements

The author wishes to express his sincere thanks to Prof. Franco Giannessi, Editor-in-Chief of JOTA, and to Prof. Lionel Thibault, Associate Editor of JOTA, and to the anonymous reviewers for their support and valuable suggestions which helped to improve the final manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Istanbul Technical University, Istanbul, Turkey

    Elimhan N. Mahmudov

  2. Azerbaijan National Academy of Sciences, Institute of Control Systems, Baku, Azerbaijan

    Elimhan N. Mahmudov

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  1. Elimhan N. Mahmudov
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Correspondence to Elimhan N. Mahmudov.

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Mahmudov, E.N. Optimization of Mayer Problem with Sturm–Liouville-Type Differential Inclusions. J Optim Theory Appl 177, 345–375 (2018). https://doi.org/10.1007/s10957-018-1260-2

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  • DOI: https://doi.org/10.1007/s10957-018-1260-2

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