Proof of the maximum principle for a problem with state constraints by the v-change of time variable

Andrei V. Dmitruk; Nikolai P. Osmolovskii

doi:10.3934/dcdsb.2019090

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2019, Volume 24, Issue 5: 2189-2204. Doi: 10.3934/dcdsb.2019090

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Proof of the maximum principle for a problem with state constraints by the v-change of time variable

Andrei V. Dmitruk^1, and
Nikolai P. Osmolovskii^2,3,4,

1.
Russian Academy of Sciences, Central Economics and Mathematics Institute, Russia 117418, Moscow, Nakhimovskii prospekt, 47 and Lomonosov Moscow State University, Russia
2.
University of Technology and Humanities in Radom, Poland, 26-600 Radom, ul. Malczewskiego 20A, Poland
3.
Systems Research Institute, Polish Academy of Sciences, Warszawa
4.
Moscow State University of Civil Engineering, Russia

Received: November 30, 2017

Revised: December 31, 2018

Early access: March 2019

Published: March 2019

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Abstract

We give a new proof of the maximum principle for optimal control problems with running state constraints. The proof uses the so-called method of $ v- $change of the time variable introduced by Dubovitskii and Milyutin. In this method, the time $ t $ is considered as a new state variable satisfying the equation $ {\rm d} t/ {\rm d} \tau = v, $ where $ v(\tau)\ge0 $ is a new control and $ \tau $ a new time. Unlike the general $ v- $change with an arbitrary $ v(\tau), $ we use a piecewise constant $ v. $ Every such $ v- $change reduces the original problem to a problem in a finite dimensional space, with a continuum number of inequality constrains corresponding to the state constraints. The stationarity conditions in every new problem, being written in terms of the original time $ t, $ give a weak^* compact set of normalized tuples of Lagrange multipliers. The family of these compacta is centered and thus has a nonempty intersection. An arbitrary tuple of Lagrange multipliers belonging to the latter ensures the maximum principle.

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Mathematics Subject Classification: Primary: 49K15; Secondary: 49K27, 46N10.

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References

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