Abstract
Various types of optimality criteria and conditions, which define the set of admissible solution, are brought to the canonical form. For the problem set in this form, the optimality conditions of sliding modes are stated. It is shown that the optimality conditions emerge from these conditions in the form of the maximum principle for problems with a scalar argument and an arbitrary combination of the optimality criterion and constraints.
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Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidge, R.V., et al., Matematicheskaya teoriya optimal’nykh protsessov (Mathematical Theory of Optimal Processes), Moscow: Nauka, 1976.
Rozonoer, L.I., Pontryagin Maximum Principle in the Theory of Optimal Systems, Autom. Remote Control, 1959, no. 10, pp. 1320–1334; no. 11, pp. 1441–1458; no. 12, pp. 1561–1578.
Butkovskii, V.G., Optomal’noe upravlenie protsessami s raspredelennymi parametrami (Optimal Control of Processes with Distributed Parameters), Moscow: Nauka, 1965.
Gabasov, R. and Kirillova, F.M., Metody optimizatsii (Optimization Methods), Minsk: Belarus. Gos. Univ., 1981.
Dubovitsii, A.Ya. and Milyutin, A.A., Problems for Extremum with Constraints, Zh. Vychisl. Mat. Mat. Fiz., 1965, no. 3, pp. 22–34.
Pontryagin, L.S., Printsip maksimuma v optimal’nom upravlenii (Maximum Principle in Optimal Control), Moscow: URSS, 2004.
Arutyunov, A.V., Magaril-Il’yaev, G.G., and Tikhomirov, V.M., Printsip maksimuma Pontryagina (dokazatel’stvo i prilozheniya) (PontryaginMaximum Principle (Proof and Applications)), Moscow: Faktorial, 2006.
Tsirlin, A.M., Solution of Optimal Control Problems on the Basis of Reduction to the Simplest Isoperimetric Problem, Izd. Akad. Nauk SSSR, Tekh. Kibern., 1968, no. 6, pp. 31–46.
Tsirlin, A.M., Optimality Conditions of Averaged Problems of Mathematical Programming, Dokl. Akad. Nauk SSSR, 1992, vol. 323, no. 1, pp. 43–47.
Krotov, V.F. and Gurman, V.I., Metody i zadachi optimal’nogo upravleniya (Methods and Problems of Optimal Control), Moscow: Nauka, 1973.
Gurman, V.I., Printsip rasshireniya v ekstremal’nykh zadachakh (Extension Principle in Extremal Problems), Moscow: Fizmatlit, 1997.
Yang, L., Lectures on the Calculus of Variations and Optimal Control Theory, London: Saunders, 1969. Translated under the title Lektsii po variatsionnomy ischisleniyu i optimal’nomu upravleniyu, Moscow: Mir, 1974.
Fromovitz, S., Nonlinear Programming with Randomization, Manag. Sci., 1965, vol. 11, no. 9, pp. 831–846.
Tsirlin, A.M., Metody usrednennoi optimizatsii i ikh prilozheniya (Methods of Averaged Optimization and Their Applications), Moscow: Fizmatlit, 1997.
Tsirlin, A.M., Mean Optimization and Sliding Modes in the Problem of Optimal Control, Izv. Akad. Nauk SSSR, Tekh., Kibern., 1974, no. 2, pp. 27–33.
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Original Russian Text © A.M. Tsirlin, 2009, published in Avtomatika i Telemekhanika, 2009, No. 5, pp. 106–121.
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Tsirlin, A.M. Optimality conditions of sliding modes and the maximum principle for control problems with the scalar argument. Autom Remote Control 70, 839–854 (2009). https://doi.org/10.1134/S0005117909050117
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DOI: https://doi.org/10.1134/S0005117909050117