# A.V. Dmitruk, N.P. Osmolovskii. Variations of the  $v$-change of time  in problems with state constraints ... P. 76-92

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 $t\mapsto \tau,$ under which the original time becomes yet another state variable subject to the equation $dt/d\tau = v(\tau),$ while the additional control $v(\tau)\ge 0$ 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 by 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.

Keywords: Pontryagin maximum principle, $v$-change of time, state constraint, semi-infinite problem, Lagrange multipliers, Lebesgue-Stieltjes measure, function of bounded variation, finite-valued maximum condition, centered family of compact sets.

The paper was received by the Editorial Office on July 26, 2017.

Funding Agency: This work was supported by the Russian Foundation for Basic Research (projects no. 16-01-00585 и 17-01-00805).

Andrei Venediktovich Dmitruk, Dr. Phys.-Math. Sci., Central Economics and Mathematics Institute of the Russian Academy of Sciences, Moscow, 117418 Russia; Dept. of Optimal Control, Lomonosov Moscow State University, Moscow, 119991 Russia, e-mail: dmitruk@member.ams.org .

Nikolai Pavlovich Osmolovskii, Dr. Phys.-Math. Sci., Prof., Dept. of Applied Mathematics, Moscow State University of Civil Engineering, Moscow, 129337 Russia; of Informatics and Mathematics, University of Technology and Humanities in Radom, Poland, e-mail: osmolovski@uph.edu.pl .

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