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Variations of the v-Change of Time in Problems with State Constraints

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Abstract

For a general optimal control problem with a state constraint, we propose a proof of the maximum principle based on a v-change of the time variable t ↦ τ, under which the original time becomes yet another state variable subject to the equation dt/dτ = v(τ), while the additional control v(τ) ≥ 0 is piecewise constant and its values are arguments of the new problem. Since the state constraint generates a continuum of inequality constraints in this problem, the necessary optimality conditions involve a measure. Rewriting these conditions in terms of the original problem, we get a nonempty compact set of collections of Lagrange multipliers that fulfil the maximum principle on a finite set of values of the control and time variables corresponding to the v-change. The compact sets generated by all possible piecewise constant v-changes are partially ordered with respect to inclusion, thus forming a centered family. Taking any element of their intersection, we obtain a universal optimality condition, in which the maximum principle holds for all values of the control and time.

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Funding

This work was supported by the Russian Foundation for Basic Research (project nos. 16-01-00585 and 17-01-00805).

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Authors and Affiliations

  1. Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119991, Russia

    A. V. Dmitruk

  2. Central Economic Mathematical Institute, Russian Academy of Sciences, Moscow, 117418, Russia

    A. V. Dmitruk

  3. National Research Moscow State University of Civil Engineering, Moscow, 129337, Russia

    N. P. Osmolovskii

  4. Systems Research Institute, Polish Academy of Sciences, Warsaw, 01-447, Poland

    N. P. Osmolovskii

Authors
  1. A. V. Dmitruk
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  2. N. P. Osmolovskii
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Correspondence to A. V. Dmitruk or N. P. Osmolovskii.

Additional information

Russian Text © The Author(s), 2018, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2018, Vol. 24, No. 1, pp. 76–92.

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Dmitruk, A.V., Osmolovskii, N.P. Variations of the v-Change of Time in Problems with State Constraints. Proc. Steklov Inst. Math. 305 (Suppl 1), S49–S64 (2019). https://doi.org/10.1134/S0081543819040072

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  • DOI: https://doi.org/10.1134/S0081543819040072

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