Geodesic
Variations of $v$-change of time in an~optimal control problem with state and mixed constraints
Izvestiya. Mathematics , Tome 87 (2023) no. 4, pp. 726-767.

Voir la notice de l'article provenant de la source Math-Net.Ru

For a general optimal control problem with state and regular mixed constraints we propose a proof of the maximum principle based on the so-called $v$-change of time variable $t \mapsto \tau$, under which the original time becomes an additional state variable subject to the equation $dt/d\tau = v(\tau)$, while the additional control variable $v(\tau)\geqslant 0$ is piecewise constant, and its values become arguments of the new problem.
Keywords: state and mixed constraints, positively linearly independent vectors, $v$-change of time, stationarity conditions, Lagrange multipliers, functional on $L_\infty$, weak* compactness, maximum principle.
Mots-clés : Lebesgue–Stieltjes measure 
@article{IM2_2023_87_4_a2,
     author = {A. V. Dmitruk},
     title = {Variations of $v$-change of time in an~optimal control problem with state and mixed constraints},
     journal = {Izvestiya. Mathematics },
     pages = {726--767},
     publisher = {mathdoc},
     volume = {87},
     number = {4},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2023_87_4_a2/}
}
TY  - JOUR
AU  - A. V. Dmitruk
TI  - Variations of $v$-change of time in an~optimal control problem with state and mixed constraints
JO  - Izvestiya. Mathematics 
PY  - 2023
SP  - 726
EP  - 767
VL  - 87
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2023_87_4_a2/
LA  - en
ID  - IM2_2023_87_4_a2
ER  - 
%0 Journal Article
%A A. V. Dmitruk
%T Variations of $v$-change of time in an~optimal control problem with state and mixed constraints
%J Izvestiya. Mathematics 
%D 2023
%P 726-767
%V 87
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2023_87_4_a2/
%G en
%F IM2_2023_87_4_a2
A. V. Dmitruk. Variations of $v$-change of time in an~optimal control problem with state and mixed constraints. Izvestiya. Mathematics , Tome 87 (2023) no. 4, pp. 726-767. http://geodesic.mathdoc.fr/item/IM2_2023_87_4_a2/

[1] R. V. Gamkrelidze, “Optimal control processes for bounded phase coordinates”, Izv. Akad. Nauk SSSR Ser. Mat., 24:3 (1960), 315–356 (Russian) | Zbl

[2] M. R. Hestenes, Calculus of variations and optimal control theory, John Wiley Sons, Inc., New York–London–Sydney, 1966 | MR | Zbl

[3] R. F. Hartl, S. P. Sethi, and R. G. Vickson, “A survey of the maximum principles for optimal control problems with state constraints”, SIAM Rev., 37:2 (1995), 181–218 | DOI | MR | Zbl

[4] A. Dmitruk and I. Samylovskiy, “On the relation between two approaches to necessary optimality conditions in problems with state constraints”, J. Optim. Theory Appl., 173:2 (2017), 391–420 | DOI | MR | Zbl

[5] A. Ya. Dubovitskii and A. A. Milyutin, “Extremum problems in the presence of restrictions”, Zh. Vychisl. Mat. Mat. Fiz., 5:3 (1965), 395–453 ; English transl. U.S.S.R. Comput. Math. Math. Phys., 5:3 (1965), 1–80 | MR | Zbl | DOI

[6] A. A. Milyutin, “Maximum principle in a regular problem of optimal control”, Necessary condition in optimal control, Chaps. 1–5, Nauka, Moscow, 1990 (Russian) | MR | Zbl

[7] A. A. Milyutin, A. V. Dmitruk, and N. P. Osmolovskii, Maximum princiiple in optimal control, Moscow State Univ., Faculty of Mech. and Math., Moscow, 2004 (Russian) https://kafedra-opu.ru/node/139

[8] A. A. Milyutin, “General schemes of necessary conditions for extrema and problems of optimal control”, Uspekhi Mat. Nauk, 25:5(155) (1970), 110–116 ; English transl. Russian Math. Surveys, 25:5 (1970), 109–115 | MR | Zbl | DOI

[9] A. Ya. Dubovitskii and A. A. Milyutin, “Necessary conditions for a weak extremum in optimal control problems with mixed constraints of the inequality type”, Zh. Vychisl. Mat. Mat. Fiz., 8:4 (1968), 725–779 ; English transl. U.S.S.R. Comput. Math. Math. Phys., 8:4 (1968), 24–98 | MR | Zbl | DOI

[10] A. Ya. Dubovitskii and A. A. Milyutin, Necessary weak extremum conditions in a general optimal control problem, Nauka, In-t Khim. Fiz. AN SSSR, Moscow, 1971 (Russian)

[11] A. Ya. Dubovitskii and A. A. Milyutin, “Maximum principle theory”, Methods of the theory of extremal problems in economics, ed. V. L. Levin, Nauka, Moscow, 1981, 6–47 (Russian) | MR | Zbl

[12] K. Makowski and L. W. Neustadt, “Optimal control problems with mixed control-phase variable equality and inequality constraints”, SIAM J. Control, 12:2 (1974), 184–228 | DOI | MR | Zbl

[13] A. M. Ter-Krikorov, “Convex programming in a space adjoint to a Banach space and convex optimal control problems with phase constraints”, Zh. Vychisl. Mat. Mat. Fiz., 16:2 (1976), 351–358 ; English transl. U.S.S.R. Comput. Math. Math. Phys., 16:2 (1976), 68–75 | MR | Zbl | DOI

[14] A. N. Dyukalov and A. Y. Ilyutovich, “Indicator of optimality in nonlinear control problems with mixed constraints. I”, Avtomat. i Telemekh., 1977, no. 3, 96–106 ; II, no. 5, 11–20 ; English transl. Autom. Remote Control, 38:3 (1977), 381–389; “Features of optimality in nonlinear problems of optimal control with mixed constraints. II”:5, 620–628 | MR | Zbl | MR | Zbl

[15] A. V. Dmitruk, “Maximum principle for the general optimal control problem with phase and regular mixed constraints”, Optimality of control dynamical systems, 14, Nauka, Vsesoyuznyi Nauchno Issled. Inst. Sist. Issled., Moscow, 1990, 26–42 ; English transl. Comput. Math. and Modeling, 4 (1993), 364–377 | Zbl | DOI | MR

[16] R. V. Gamkrelidze, “Optimal sliding states”, Dokl. Akad. Nauk SSSR, 143:6 (1962), 1243–1245 ; English transl. Soviet Math. Dokl., 3 (1962), 559–562 | MR | Zbl

[17] E. N. Devdariani and Yu. S. Ledyaev, “Maximum principle for implicit control systems”, Appl. Math. Optim., 40:1 (1999), 79–103 | DOI | MR | Zbl

[18] M. R. de Pinho and J. F. Rosenblueth, “Necessary conditions for constrained problems under Mangasarian–Fromowitz conditions”, SIAM J. Control Optim., 47:1 (2008), 535–552 | DOI | MR | Zbl

[19] F. Clarke and M. R. de Pinho, “Optimal control problems with mixed constraints”, SIAM J. Control Optim., 48:7 (2010), 4500–4524 | DOI | MR | Zbl

[20] H. A. Biswas and M. d. R. de Pinho, “A maximum principle for optimal control problems with state and mixed constraints”, ESAIM Control Optim. Calc. Var., 21:4 (2015), 939–957 | DOI | MR | Zbl

[21] A. Boccia, M. D. R. de Pinho, and R. B. Vinter, “Optimal control problems with mixed and pure state constraints”, SIAM J. Control Optim., 54:6 (2016), 3061–3083 | DOI | MR | Zbl

[22] An Li and J. J. Ye, “Necessary optimality conditions for optimal control problems with nonsmooth mixed state and control constraints”, Set-Valued Var. Anal., 24:3 (2016), 449–470 | DOI | MR | Zbl

[23] R. Andreani, V. A. de Oliveira, J. T. Pereira, and G. N. Silva, “A weak maximum principle for optimal control problems with mixed constraints under a constant rank condition”, IMA J. Math. Control Inform., 37:3 (2020), 1021–1047 | DOI | MR | Zbl

[24] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The mathematical theory of optimal processes, Fizmatgiz, Moscow, 1961 ; 2nd ed., Nauka, Moscow, 1969 ; English transl. of the first edition Intersci. Publ. John Wiley Sons, Inc., New York–London, 1962 | MR | Zbl | Zbl | MR | Zbl

[25] A. V. Dmitruk and N. P. Osmolovskii, “On the proof of Pontryagin's maximum principle by means of needle variations”, Fundam. Prikl. Mat., 19:5 (2014), 49–73 ; English transl. J. Math. Sci. (N.Y.), 218:5 (2016), 581–598 | MR | Zbl | DOI

[26] G. G. Magaril-Il'yaev, “The Pontryagin maximum principle. Ab ovo usque ad mala”, Optimal control, Collected papers. In commemoration of the 105th anniversary of Academician Lev Semenovich Pontryagin, Trudy Mat. Inst. Steklova, 291, MAIK “Nauka/Interperiodica”, Moscow, 2015, 215–230 ; English transl. G. G. Magaril-Il'yaev, 291, 2015, 203–218 | DOI | MR | Zbl | DOI

[27] A. V. Dmitruk and N. P. Osmolovskii, “Variations of the $v$-change of time in problems with state constraints”, Trudy Inst. Mat. i Mekh. UrO RAN, 24, no. 1, 2018, 76–92 ; English transl. Proc. Steklov Inst. Math. (Suppl.), 305, suppl. 1 (2019), S49–S64 | DOI | MR | Zbl | DOI

[28] A. V. Dmitruk and N. P. Osmolovskii, “Proof of the maximum principle for a problem with state constraints by the $v$-change of time variable”, Discrete Contin. Dyn. Syst. Ser. B, 24:5 (2019), 2189–2204 | DOI | MR | Zbl

[29] A. V. Dmitruk, “Approximation theorem for a nonlinear control system with sliding modes”, Dynamical systems and optimization, Collected papers. Dedicated to the 70th birthday of academician Dmitrii Viktorovich Anosov, Trudy Mat. Inst. Steklova, 256, Nauka, MAIK Nauka/Inteperiodika, Moscow, 2007, 102–114 ; English transl. Proc. Steklov Inst. Math., 256 (2007), 92–104 | MR | Zbl | DOI

[30] A. A. Milyutin, The maximum principle in the general problem of optimal control, Fizmatlit, Moscow, 2001 (Russian) | Zbl

[31] A. V. Dmitruk, “On the development of Pontryagin's maximum principle in the works of A. Ya. Dubovitskii and A. A. Milyutin”, Control Cybernet., 38:4A (2009), 923–957 | MR | Zbl

[32] “Necessary extremum conditions (Lagrange principle)”, Optimal control, Chap. 3, eds. V. M. Tikhomirov and N. P. Osmolovskii, MCCME, Moscow, 2008, 89–122 (Russian)

[33] A. A. Milyutin and N. P. Osmolovskii, “First order conditions”, Calculus of variations and optimal control, Transl. from the Russian manuscript, Transl. Math. Monogr., 180, Amer. Math. Soc., Providence, RI, 1998 | DOI | MR | Zbl

[34] A. V. Dmitruk and A. M. Kaganovich, “Maximum principle for optimal control problems with intermediate constraints”, Nonlinear dynamics and control, 6, Fizmatlit, Moscow, 2008, 101–136 ; English transl. Comput. Math. Model., 22:2 (2011), 180–215 | MR | Zbl | DOI

[35] 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. Control Optim., 52:6 (2014), 3437–3462 | DOI | MR | Zbl

[36] A. Dmitruk and N. Osmolovskii, “A general Lagrange multipliers theorem”, 2017 Constructive nonsmooth analysis and related topics (dedicated to the memory of V. F. Demyanov), CNSA-2017 (St. Petersburg 2017), IEEE, 2017, 82–84 | DOI

[37] A. V. Dmitruk and N. P. Osmolovskii, “A general Lagrange multipliers theorem and related questions”, Control systems and mathematical methods in economics, Lecture Notes in Econom. and Math. Systems, 687, Springer, Cham, 2018, 165–194 | DOI | MR | Zbl