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Discrete Approximations and Optimality Conditions for Controlled Free-Time Sweeping Processes

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Abstract

The paper is devoted to the study of a new class of optimal control problems governed by discontinuous constrained differential inclusions of the sweeping type involving the duration of the dynamic process into optimization. We develop a novel version of the method of discrete approximations of its own qualitative and numerical values with establishing its well-posedness and strong convergence to optimal solutions of the controlled sweeping process. Using advanced tools of first-order and second-order variational analysis and generalized differentiation allows us to derive new necessary conditions for optimal solutions of the discrete-time problems and then, by passing to the limit in the discretization procedure, for designated local minimizers in the original problem of sweeping optimal control. The obtained results are illustrated by a numerical example.

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Notes

  1. Here we mean the union of the density points of E such that the normal cone to C(t) is active along E.

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Authors and Affiliations

  1. Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova, 35121, Padova, Italy

    Giovanni Colombo

  2. G.N.A.M.P.A. of Istituto Nazionale di Alta Matematica “Francesco Severi”, Piazzale Aldo Moro 5, 00185, Roma, Italy

    Giovanni Colombo

  3. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Boris S. Mordukhovich & Trang Nguyen

  4. Department of Mathematics and Statistics, San Diego State University, San Diego, CA, 92182, USA

    Dao Nguyen

Authors
  1. Giovanni Colombo
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  2. Boris S. Mordukhovich
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  3. Dao Nguyen
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  4. Trang Nguyen
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Correspondence to Boris S. Mordukhovich.

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The authors declare that the presented results are new, and there is no any conflict of interest.

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The research of G. Colombo was partly supported by the project funded by the EuropeanUnion - NextGenerationEU under the National Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 1.1 - Call PRIN 2022 No. 104 of February 2, 2022 of Italian Ministry of University and Research; Project 2022238YY5 (subject area: PE - Physical Sciences and Engineering) “Optimal control problems: analysis, approximation and applications”.

Research of B. S. Mordukhovich was partially supported by the USA National Science Foundation under grants DMS-1808978 and DMS-2204519, by the Australian Research Council under grant DP-190100555, and by Project 111 of China under grant D21024.

Research of D. Nguyen was supported by the AMS-Simon Foundation.

Research of T. Nguyen was partially supported by the USA National Science Foundation under grant DMS-1808978 and DMS-2204519.

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Colombo, G., Mordukhovich, B.S., Nguyen, D. et al. Discrete Approximations and Optimality Conditions for Controlled Free-Time Sweeping Processes. Appl Math Optim 89, 40 (2024). https://doi.org/10.1007/s00245-024-10108-7

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