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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 |
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.
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.
|
[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.
|
[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. |
[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. |
[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). |
[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. |
[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.
|
[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. |
[10] |
I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Lecture Notes in Economics and Mathematical Systems, 1972. |
[11] |
A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, North-Holland Publishing Company, Amsterdam, 1979, Russian Edition: Nauka, Moscow, 1974. |
[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. |
[13] |
A. A. Milyutin,
General schemes of necessary conditions for extrema and problems of optimal control, Russian Mathematical Surveys, 25 (1970), 110-116.
|
[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.
![]() |
[17] |
R. Vinter, Optimal Control, Birkhauser, Boston, 2000. |
[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. |
[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/). |
[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 |
show all references
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.
|
[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.
|
[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. |
[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. |
[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). |
[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. |
[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.
|
[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. |
[10] |
I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Lecture Notes in Economics and Mathematical Systems, 1972. |
[11] |
A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, North-Holland Publishing Company, Amsterdam, 1979, Russian Edition: Nauka, Moscow, 1974. |
[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. |
[13] |
A. A. Milyutin,
General schemes of necessary conditions for extrema and problems of optimal control, Russian Mathematical Surveys, 25 (1970), 110-116.
|
[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.
![]() |
[17] |
R. Vinter, Optimal Control, Birkhauser, Boston, 2000. |
[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. |
[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/). |
[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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