Analysis of sufficient optimality conditions with a set of Lyapunov type functions

V. A. Dykhta

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 5, pp. 66-75

Résumé

For the classical optimal control problem with general endpoints constraints new variants of the Krotov–Carathéodory type global optimality conditions and sufficient conditions of the so-called Hamilton–Jacobi canonical optimality theory are proposed and compared. These sufficient optimality conditions are obtained and formulated by using some support set of nonsmooth Lyapunov-type functions, which are strongly monotone with respect to the control dynamic system. It is proved that the sufficient optimality conditions corresponding to the canonical theory are more efficient. Some properties of Lyapunov functions from the support set are investigated.

Détail

Keywords: sufficient optimality conditions, Lyapunov functions, Hamilton–Jacobi inequalities, support set.

Bibliographie

[1] Klark F., Optimizatsiya i negladkii analiz, Nauka, M., 1988, 280 pp. | MR | Zbl

[2] Yang L., Lektsii po variatsionnomu ischisleniyu i teorii optimalnogo upravleniya, Mir, M., 1974, 488 pp. | MR

[3] Krotov V. F., Gurman V. I., Metody i zadachi optimalnogo upravleniya, Nauka, M., 1973, 448 pp. | MR | Zbl

[4] Clarke F. H. et al. (eds.), Nonsmooth Analysis and Control Theory, Grad. Texts in Math., 178, Springer-Verlag, N.Y., 1998, 276 pp. | MR | Zbl

[5] Krotov V. F., Global Methods in Optimal Control Theory, Monographs and Textbooks in Pure and Applied Mathematics, 195, Marcel Dekker, N.Y., 1996, 384 pp. | MR | Zbl

[6] Bardi M., Capuzzo-Dolcetta I., Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations, Birkhauser, Boston, 1997, 570 pp. | MR | Zbl

[7] Dykhta V. A., “Nekotorye prilozheniya neravenstv Gamiltona–Yakobi v optimalnom upravlenii”, Izv. IGU (Matematika), 2 (2009), 15–28

[8] Milyutin A. A., “Calculus of variations and optimal control”, Proc. Internat. Conf. on the Calculus of Variations and Related Topics (Haifa,), Chapman and Hall/CRC Research Notes in Mathematics Series, 411, 2000, 159–172 | MR | Zbl

[9] Milyutin A. A., Osmolovskii N. P., Calculus of Variations and Optimal Control, Amer. Math. Soc., Providence, Rhode Island, 1998, 372 pp. | MR | Zbl

[10] Dykhta V. A., “Printsip rasshireniya v kachestvennoi teorii upravleniya”, Metody resheniya zadach teorii upravleniya na osnove printsipa rasshireniya, eds. V. I. Gurman, G. N. Konstantinov, Nauka, Novosibirsk, 1990, 190 pp. | MR | Zbl

[11] Dykhta V. A., “Neravenstvo Lyapunova–Krotova i dostatochnye usloviya v optimalnom upravlenii”, Itogi nauki i tekhniki. Sovr. matematika i ee prilozheniya, 110, 2006, 76–108 | MR | Zbl

[12] Subbotin A. I., Obobschennye resheniya uravnenii v chastnykh proizvodnykh pervogo poryadka. Perspektivy dinamicheskoi optimizatsii, In-t komp. issledovanii, M.–Izhevsk, 2003, 336 pp.

[13] Subbotin A. I., Minimaksnye neravenstva i uravneniya Gamiltona–Yakobi, Nauka, M., 1991, 216 pp. | MR | Zbl

[14] Vinter R. B., Optimal Control, Birkhäuser, Boston, 2000, 504 pp. | MR | Zbl

[15] Clarke F. H., Nour C., “Nonconvex duality in optimal control”, SIAM J. Control Optim., 43 (2005), 2036–2048 | DOI | MR | Zbl

[16] Vinter R. B., “Convex duality and nonlinear optimal control”, SIAM J. Control Optim., 31 (1993), 518–538 | DOI | MR | Zbl

[17] Bacciotti A., Rosier L., Lyapunov functions and stability in control theory, Springer-Verlag, Berlin–Heidelberg, 2005, 235 pp. | MR | Zbl

[18] Matrosov V. M., Metod vektornykh funktsii Lyapunova: analiz dinamicheskikh svoistv nelineinykh sistem, Fizmatlit, M., 2001, 384 pp.

[19] Gurman V. I., Printsip rasshireniya v zadachakh upravleniya, 2-e izd., pererab. i dop., Nauka, Fizmatlit, M., 1997, 288 pp. | MR | Zbl

[20] Arguchintsev A. V., Dykhta V. A., Srochko V. A., “Optimalnoe upravlenie: nelokalnye usloviya, vychislitelnye metody i variatsionnyi printsip maksimuma”, Izv. vuzov. Matematika, 2009, no. 1, 3–43 | MR | Zbl

[21] Khrustalev M. M., “Tochnoe opisanie mnozhestv dostizhimosti i uslovie globalnoi optimalnosti dinamicheskikh sistem. I. Otsenki i tochnoe opisanie mnozhestv dostizhimosti i upravlyaemosti”, Avtomatika i telemekhanika, 1988, no. 5, 62–71 | MR

[22] Khrustalev M. M., “Tochnoe opisanie mnozhestv dostizhimosti i uslovie globalnoi optimalnosti dinamicheskikh sistem. II. Usloviya globalnoi optimalnosti”, Avtomatika i telemekhanika, 1988, no. 7, 70–80 | MR

[23] Dmitruk A. V., “Quadratic order conditions of a local minimum for singular extremals in a general optimal control problem”, Proc. of Symp. Pure Math. Differential Geometry and Control, 64, eds. G. Fereira et al., Amer. Math. Soc., 1998, 163–198 | MR

[24] Ekeland I., “Nonconvex minimization problems”, Bull. Amer. Math. Soc., 1:3 (1979), 443–474 | DOI | MR | Zbl

[25] Subbotina N. N., “Printsip maksimuma i subdifferentsial funktsii tseny”, Problemy upravleniya i teorii informatsii, 18:3 (1989), 151–160 | MR | Zbl

[26] Subbotina N. N., “Metod dinamicheskogo programmirovaniya dlya klassa lokalno-lipshitsevykh funktsii”, Dokl. RAN, 389:2 (2003), 169–172 | Zbl

[27] Mordukhovich B. S., Variational Analysis and Generalized Differentiation I, Fundamental Principles of Mathematical Sciences, 330, Springer, Berlin, 2006, 611 pp. ; II, Fundamental Principles of Mathematical Sciences, 331, 580 pp. | MR | Zbl | MR

[28] Borschevskii M. Z., “Zadacha dinamicheskogo ispolzovaniya istoschaemogo resursa”, Voprosy prikladnoi matematiki, SEI SO AN SSSR, Irkutsk, 1977, 120–131