Geodesic
Dynamic method of multipliers in terminal control
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 5, pp. 776-797 
A. S. Antipin; O. O. Vasilieva. Dynamic method of multipliers in terminal control. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 5, pp. 776-797. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_5_a3/
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Voir la notice de l'article provenant de la source Math-Net.Ru

A method for solving the terminal control problem with a fixed time interval and fixed initial conditions is proposed. The solution to the boundary value problem posed at the right end of the time interval determines the terminal conditions. This boundary value problem is a finite-dimensional convex programming problem. The dynamics of the terminal control problem is described by a linear controllable system of differential equations. This system is interpreted as a conventional system of linear equality constraints. Then the terminal control problem can be regarded as a dynamic convex programming problem posed in an infinite-dimensional functional Hilbert space. In this paper, the functional problem is treated as a saddle-point problem rather than optimization problem. Accordingly, a saddle-point approach to solving the problem is proposed. This approach is based on maximizing the dual function generated by the modified Lagrangian function of the convex programming problem posed in the functional space. The convergence of the proposed methods is also proved in the functional space. This convergence has the additional property of being monotone in norm with respect to controls, phase trajectories, adjoint functions, as well as finite-dimensional terminal variables.

[1] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Fizmatlit, M., 2009

[2] Vasilev F. P., Metody optimizatsii, v. 1, 2, Izd-vo MTsNMO, M., 2011

[3] Antipin A. S., “Ravnovesnoe programmirovanie modeli i metody resheniya”, Izvestiya IGU. Ser. “Matematika”, 2:1 (2009), 8–36

[4] Antipin A. S., “Two-person game with Nash equilibrium in optimal control problems”, Optim. Lett., 6:7 (2012), 1349–1378 | MR | Zbl

[5] Antipin A. S., Khoroshilova E. V., “Lineinoe programmirovanie i dinamika”, Tr. In-ta matem. i mekhan. UrO RAN, 19, no. 2, 2013, 7–25

[6] Antipin A. S., Khoroshilova E. V., “Optimalnoe upravlenie so svyazannymi nachalnymi i terminalnymi usloviyami”, Tr. In-ta matem. i mekhan. UrO RAN, 20, no. 2, 2014, 13–28

[7] Antipin A. S., “Terminalnoe upravlenie kraevymi modelyami”, Zh. vychisl. matem. i matem. fiz., 54:2 (2014), 257–285 | MR | Zbl

[8] Antipin A. S., Vasilieva O. O., “Augmented Lagrangrian method for optimal control problems”, Analysis, Modelling, Optimization, and Numerical Tecniques, Springer Proc. in Mathematics Statistics, 121, eds. G. Olivar Tost, O. Vasilieva, 2015, 1–36 | Zbl

[9] Antipin A. S., “O metode vypuklogo programmirovaniya, ispolzuyuschem simmetricheskuyu modifikatsiyu funktsii Lagranzha”, Ekonomika i matem. metody, XII:6 (1976), 1164–1173 | Zbl

[10] Antipin A. S., “Ob odnom metode otyskaniya sedlovoi tochki modifitsirovannoi funktsii Lagranzha”, Ekonomika i mat. metody, 13:3 (1977), 560–565 | MR | Zbl

[11] Golshtein E. G., Tretyakov N. V., Modifitsirovannye funktsii Lagranzha. Teoriya i metody optimizatsii, Nauka, M., 1989 | MR

[12] Polyak B. T., Vvedenie v optimizatsiyu, Nauka, M., 1983 | MR

[13] Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mischenko E. F., Matematicheskaya teoriya optimalnykh protsessov, Nauka, M., 1976

[14] Antipin A. S., “Ravnovesnoe programmirovanie: metody gradientnogo tipa”, Avtomatika i telemekhanika, 1997, no. 8, 125–137 | MR | Zbl

[15] Antipin A. S., “Ravnovesnoe programmirovanie: proksimalnye metody”, Zh. vychisl. matem. i matem. fiz., 37:11 (1997), 1327–1339 | MR | Zbl

[16] Antipin A. S., “Extra-proximal methods for solving two-person nonzero-sum games”, Math. Program. Ser. B, 120:1 (2009), 147–177 | MR | Zbl

[17] Rockafellar R. T., “Augmented Lagrangians and applications of the proximal point algorithm in convex programming”, Math. Oper. Res., 1:2 (1976), 97–116 | MR | Zbl

[18] Hager W. W., “Multiplier methods for nonlinear optimal control”, SIAM J. Numer. Anal., 27:4 (1990), 1061–1080 | MR | Zbl

[19] Vasilieva O. O., “The search of equilibrium strategies for controlled boundary value problem”, Asian J. of Control., 3:1 (2001), 50–56

[20] Vasileva O. O., “Poisk ravnovesnykh upravlenii dlya upravlyaemykh kraevykh zadach”, Izv. IGU. Ser. Matematika, 1:1 (2007), 70–85

[21] Vasilieva O. O., Vasil'ev O. V., “On the search for equilibrium controls in an $m$-person differential game”, Russian Math., 44:12 (2000), 7–12 | MR

[22] Lyusternik L. A., Sobolev V. I., Elementy funktsionalnogo analiza, Nauka, M., 1965 | MR

[23] Rockafellar R. T., Wets R. J.-B., Variational analysis, Grundlehren der Mathematischen Wissenschaften, 317, Springer-Verlag, Berlin, 1998 | MR | Zbl