# American Institute of Mathematical Sciences

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:
 [1] 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 [2] 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 [3] 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 [4] 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 [5] 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 [6] 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 [7] 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 [8] 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 [9] 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 [10] I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Lecture Notes in Economics and Mathematical Systems, 1972.  Google Scholar [11] A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, North-Holland Publishing Company, Amsterdam, 1979, Russian Edition: Nauka, Moscow, 1974.  Google Scholar [12] 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 [13] A. A. Milyutin, General schemes of necessary conditions for extrema and problems of optimal control, Russian Mathematical Surveys, 25 (1970), 110-116.   Google Scholar [14] A. A. Milyutin, Maximum Principle in the General Optimal Control Problem [in Russian], Fizmatlit, Moscow, 2001. Google Scholar [15] A. A. Milyutin, A. V. Dmitruk and N. P. Osmolovsky, Maximum Principle in Optimal Control, Moscow State University, Moscow, 2004 (in Russian). Google Scholar [16] 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 [17] R. Vinter, Optimal Control, Birkhauser, Boston, 2000.  Google Scholar [18] 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 [19] A. Ya. Dubovitskii and A. A. Milyutin, Translation of Euler's equations, USSR Comput. Math. and Math. Physics, 9 (1969), 37-64.   Google Scholar [20] 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 [21] 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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##### References:
 [1] 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 [2] 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 [3] 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 [4] 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 [5] 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 [6] 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 [7] 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 [8] 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 [9] 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 [10] I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Lecture Notes in Economics and Mathematical Systems, 1972.  Google Scholar [11] A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, North-Holland Publishing Company, Amsterdam, 1979, Russian Edition: Nauka, Moscow, 1974.  Google Scholar [12] 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 [13] A. A. Milyutin, General schemes of necessary conditions for extrema and problems of optimal control, Russian Mathematical Surveys, 25 (1970), 110-116.   Google Scholar [14] A. A. Milyutin, Maximum Principle in the General Optimal Control Problem [in Russian], Fizmatlit, Moscow, 2001. Google Scholar [15] A. A. Milyutin, A. V. Dmitruk and N. P. Osmolovsky, Maximum Principle in Optimal Control, Moscow State University, Moscow, 2004 (in Russian). Google Scholar [16] 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 [17] R. Vinter, Optimal Control, Birkhauser, Boston, 2000.  Google Scholar [18] 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 [19] A. Ya. Dubovitskii and A. A. Milyutin, Translation of Euler's equations, USSR Comput. Math. and Math. Physics, 9 (1969), 37-64.   Google Scholar [20] 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 [21] 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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