    May  2019, 24(5): 2189-2204. doi: 10.3934/dcdsb.2019090

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

 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  December 2017 Revised  January 2019 Published  March 2019

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.

Citation: Andrei V. Dmitruk, Nikolai P. Osmolovskii. Proof of the maximum principle for a problem with state constraints by the v-change of time variable. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2189-2204. doi: 10.3934/dcdsb.2019090
##### References:
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##### References:
  L. Bourdin, Note on Pontryagin maximum principle with running state constraints and smooth dynamics - Proof based on the Ekeland variational principle, arXiv: 1604.04051 [math.OC]. Google Scholar  A. V. Dmitruk, A. A. Milyutin and N. P. Osmolovsky, Lyusternik's theorem and the theory of extrema, Russian Math. Surveys, 35 (1980), 11-46. Google Scholar  A. V. Dmitruk, On the development of Pontryagin's Maximum principle in the works of A.Ya. Dubovitskii and A.A. Milyutin, Control and Cybernetics, 38 (2009), 923-957. Google Scholar  A. V. Dmitruk and N. P. Osmolovskii, Necessary conditions for a weak minimum in optimal control problems with integral equations subject to state and mixed constraints, SIAM J. on Control and Optimization, 52 (2014), 3437-3462.  doi: 10.1137/130921465.  Google Scholar  A. V. Dmitruk and N. P. Osmolovskii, On the proof of Pontryagin's Maximum principle by means of needle variations, Journal of Mathematical Sciences, 218 (2016), 581-598.  doi: 10.1007/s10958-016-3044-2.  Google Scholar  A. V. Dmitruk and N. P. Osmolovskii, Variations of the type of $v-$change of time in problems with state constraints, Proc. of the Institute of Mathematics and Mechanics, the Ural Branch of Russian Academy of Sciences, 24 (2018), 76-92 (in Russian). Google Scholar  A. V. Dmitruk and N. P. Osmolovskii, A General Lagrange Multipliers Theorem, Constructive Nonsmooth Analysis and Related Topics (CNSA-2017), IEEE Xplore Digital Library, 2017. doi: 10.1109/CNSA.2017.7973951. Google Scholar  A. V. Dmitruk and N. P. Osmolovskii., A General Lagrange Multipliers Theorem and Related Questions, Lecture Notes in Economics and Math. Systems, 687 (2018), 165-194. Google Scholar  I. Ekeland, Nonconvex minimization problems, Bull. of American Math. Society (New Series), 1 (1979), 443-474.  doi: 10.1090/S0273-0979-1979-14595-6.  Google Scholar  I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Lecture Notes in Economics and Mathematical Systems, 1972. Google Scholar  A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, North-Holland Publishing Company, Amsterdam, 1979, Russian Edition: Nauka, Moscow, 1974. Google Scholar  A. N. Kolmogorov and S. V. Fomin., Elements of the Theory of Functions and Functional Analysis, Dover Books on Mathematics, 1999; Russian 4th Edition: Nauka, Moscow, 1976. Google Scholar  A. A. Milyutin, General schemes of necessary conditions for extrema and problems of optimal control, Russian Mathematical Surveys, 25 (1970), 110-116. Google Scholar  A. A. Milyutin, Maximum Principle in the General Optimal Control Problem [in Russian], Fizmatlit, Moscow, 2001. Google Scholar  A. A. Milyutin, A. V. Dmitruk and N. P. Osmolovsky, Maximum Principle in Optimal Control, Moscow State University, Moscow, 2004 (in Russian). Google Scholar  L. S. Pontryagin, V. G. Boltyanski, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Fitzmatgiz, Moscow; English translation: Pergamon Press, New York, 1964. Google Scholar  R. Vinter, Optimal Control, Birkhauser, Boston, 2000. Google Scholar  A. Ya. Dubovitskii and A. A. Milyutin, Extremum problems in the presence of restrictions, USSR Comput. Math. and Math. Physics, 5 (1965), 1-80.  doi: 10.1016/0041-5553(65)90148-5. Google Scholar  A. Ya. Dubovitskii and A. A. Milyutin, Translation of Euler's equations, USSR Comput. Math. and Math. Physics, 9 (1969), 37-64.   Google Scholar  A. Ya. Dubovitskii and A. A. Milyutin, Theory of the maximum principle, Methods of the Theory Of Extremal Problems in Economics (V.L. Levin ed.), Nauka, Moscow, (1981), 6-47 (in Russian, see http://www.milyutin.ru/). Google Scholar  Optimal'noe upravlenie [Optimal Control], (N.P. Osmolovskii and V.M. Tikhomirov eds.), Moscow Center for Continuous Mathematical Education (MCCME), Moscow, Russia, 2008 (in Russian). Google Scholar
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