On the maximum principle when the endpoints of the optimal trajectory lie on the state boundary

• Autorzy: Dmitruk Andrei V. , Osmolovskii Nikolai P.

Identyfikatory

DOI

10.2478/candc-2024-0005

• Języki publikacji: EN

Abstrakty

EN

We consider an optimal control problem in the Mayer form with an autonomous control system, a bounded control set U, endpoint equality constraints, and one pointwise state inequality constraint. We analyze the case when the starting point of the optimal trajectory belongs to the state boundary. A nontrivial maximum principle was obtained for this case by A.Ya. Dubovitskii and V.A. Dubovitskii about 40 years ago, but the proof was written in an extremely condensed form. Here we offer a new proof of their result.

Słowa kluczowe

EN

optimal control; normed space; state constraint; Pontryagin function; Lebesgue–Stieltjes measure; singular measure; costate equation

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2024
• Tom: Vol. 53, No. 1
• Strony: 77--107
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Dmitruk Andrei V.

• Russian Academy of Sciences, Central Economics and Mathematics Institute, Moscow, Russia

autor

Osmolovskii Nikolai P.

• Systems Research Institute, Polish Academy of Science, Warszawa, Poland

Bibliografia

• Arutyunov, A. and Tynyanskiy, N. (1984) The maximum principle in a problem with phase constraints. Izv. Akad. Nauk SSSR Tekhn. Kibernet., 0-060-68, 235.
• Dubovitskii, A.Ya. and Dubovitskii, V.A. (1985) Necessary conditions for a strong minimum in optimal control problems with degeneracy of the end-point and phase constraints. Russian Math. Surveys, 40, 2, 209–210.
• Dubovitskii, A.Ya. and Dubovitskii, V.A. (1987) The maximum principle in regular optimal control problems where the ends of the phase path are on the boundary of the phase constraint. Avtomatika i Telemekhanika, 12, 25–33 (in Russian).
• Dubovitskii, A.Ya. and Dubovitskii, V.A. (1995) A criterion for the existence of a meaningful maximum principle in a problem with phase constraints. Diff. Equat., 31, 10, 1595–1602.
• Dubovitskii, A. Ya. and Milyutin, A.A. (1965) Extremum problems in the presence of restrictions. USSR Comput. Math. and Math. Phys., 5, 3, 1–80.
• Dmitruk, A. V. and Osmolovskii, N. P. (2018) 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, 76–92 (in Russian).
• Dmitruk, A. V. and Osmolovskii, N. P. (2019) Proof of the maximum principle for a problem with state constraints by the v-change of time variable. Discrete and Continuous Dynamical Systems, Ser. B, 24, 5, 2189–2204. doi:10.3934/dcdsb.2019090.
• Dmitruk, A. V. (2009) On the development of Pontryagin’s Maximum principle in the works of A.Ya. Dubovitskii and A.A. Milyutin. Control and Cybernetics 38, 4a, 923–958.
• Dmitruk, A. V. and Osmolovskii, N. P. (2017) A General Lagrange Multipliers Theorem. Constructive Nonsmooth Analysis and Related Topics (CNSA-2017). IEEE Xplore Digital Library. doi:10.1109/CNSA.2017. 7973951.
• Dmitruk, A. V. and Osmolovskii, N. P. (2018) A General Lagrange Multipliers Theorem and Related Questions. In: G. Feichtinger et al., eds., Control Systems and Math. Methods in Economics. Lecture Notes in Economics and Mathematical Systems, 687, 165–194, Springer.
• Dmitruk, A. V. and Osmolovskii, N. P. (2020) Lagrange Multipliers Rule for a General Optimization Problem with an Infinite Number of Constraints. In: A. Vasin and F. Aleskerov, eds., Recent Advances of the Russian Operations Research Soc. Cambridge Scholars Publishing, 212–232.
• Kolmogorov, A. N. and Fomin, S. V. (1999) Elements of the Theory of Functions and Functional Analysis. Dover Books on Mathematics, Russian 4th Edition: Nauka, Moscow.
• Milyutin, A. A., Dmitruk, A. V. and Osmolovskii, N. P. (2004) Maximum Principle in Optimal Control. Moscow State University, Faculty of Mechanics and Mathematics, Moscow (in Russian), 168.