Hamilton--Jacobi inequalities in control problems for impulsive dynamical systems

V. A. Dykhta ; O. N. Samsonyuk

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and topology. II, Tome 271 (2010), pp. 93-110.

Résumé

We propose definitions of strong and weak monotonicity of Lyapunov-type functions for nonlinear impulsive dynamical systems that admit vector measures as controls and have trajectories of bounded variation. We formulate infinitesimal conditions for the strong and weak monotonicity in the form of systems of proximal Hamilton–Jacobi inequalities. As an application of strongly and weakly monotone Lyapunov-type functions, we consider estimates for integral funnels of impulsive systems as well as necessary and sufficient conditions of global optimality corresponding to the approach of the canonical Hamilton–Jacobi theory.

Bibliographie

[1] Aubin J.-P., Cellina A., Differential inclusions: Set-valued maps and viability theory, Springer, Berlin, 1984 | MR | Zbl

[2] Aubin J.-P., Frankowska H., Set-valued analysis, Birkhäuser, Boston, 1990 | MR | Zbl

[3] Clarke F.H., Ledyaev Yu.S., Stern R.J., Wolenski P.R., Nonsmooth analysis and control theory, Grad. Texts Math., 178, Springer, New York, 1998 | MR | Zbl

[4] Guseinov Kh.G., Ushakov V.N., “Silno i slabo invariantnye mnozhestva otnositelno differentsialnogo vklyucheniya, ikh proizvodnye i primenenie k zadacham upravleniya”, Dif. uravneniya, 26:11 (1990), 1888–1894 | MR

[5] Subbotin A.I., Obobschennye resheniya uravnenii v chastnykh proizvodnykh pervogo poryadka. Perspektivy dinamicheskoi optimizatsii, In-t kompyut. issled., Moskva; Izhevsk, 2003

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

[7] Bardi M., Capuzzo-Dolcetta I., Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations, Birkhäuser, Boston, 1997 | MR | Zbl

[8] Bacciotti A., Rosier L., Liapunov functions and stability in control theory, Springer, Berlin, 2005 | MR | Zbl

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

[10] Krotov V.F., Global methods in optimal control theory, Pure and Appl. Math., 195, M. Dekker, New York, 1996 | MR | Zbl

[11] Cesari L., Optimization—theory and applications: Problems with ordinary differential equations, Springer, New York, 1983 | MR | Zbl

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

[13] Vinter R., Optimal control, Birkhäuser, Boston, 2000 | MR | Zbl

[14] Milyutin A.A., Osmolovskii N.P., Calculus of variations and optimal control, Amer. Math. Soc., Providence (RI), 1998 | MR | Zbl

[15] Dykhta V.A., “Neravenstvo Lyapunova–Krotova i dostatochnye usloviya v optimalnom upravlenii”, Optimalnoe upravlenie i dinamicheskie sistemy, Itogi nauki i tekhniki. Sovr. matematika i ee pril. Tematich. obzory, 110, VINITI, M., 2006, 76–108

[16] 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

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

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

[19] Ioffe A.D., Tikhomirov V.M., Teoriya ekstremalnykh zadach, Nauka, M., 1974 | MR | Zbl

[20] Kotsiopoulos J., Vinter R.B., “Dynamic programming for free-time problems with endpoint constraints”, Math. Control Signals and Syst., 6 (1993), 180–193 | DOI | MR | Zbl

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

[22] Khrustalev M.M., “Tochnoe opisanie mnozhestv dostizhimosti i usloviya globalnoi optimalnosti dinamicheskikh sistem. I: Otsenki i tochnoe opisanie mnozhestv dostizhimosti i upravlyaemosti”, Avtomatika i telemekhanika, 1988, no. 5, 62–70 | MR | Zbl

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

[24] Clarke F.H., Ledyaev Yu.S., Subbotin A.I., “The synthesis of universal feedback pursuit strategies in differential games”, SIAM J. Control and Optim., 35 (1997), 552–561 | DOI | MR | Zbl

[25] Klark F., Ledyaev Yu.S., Subbotin A.I., “Universalnoe pozitsionnoe upravlenie i proksimalnoe pritselivanie v zadachakh upravleniya v usloviyakh vozmuscheniya i differentsialnykh igrakh”, Tr. MIAN, 224, 1999, 165–186 | MR

[26] Dykhta V.A., “Neravenstva Gamiltona–Yakobi v optimalnom upravlenii: gladkaya dvoistvennost i uluchshenie”, Vestn. Tambov. un-ta. Est. i tekhn. nauki, 15:1 (2010), 405–426

[27] Dykhta V.A., “Nekotorye prilozheniya neravenstv Gamiltona–Yakobi v optimalnom upravlenii”, Izv. Irkut. gos. un-ta. Matematika, 2:1 (2009), 183–196 | MR

[28] Zavalischin S.T., Sesekin A.N., Impulsnye protsessy: modeli i prilozheniya, Nauka, M., 1991 | MR

[29] Miller B.M., Rubinovich E.Ya., Optimizatsiya dinamicheskikh sistem s impulsnymi upravleniyami, Nauka, M., 2005

[30] Dykhta V.A., Samsonyuk O.N., Optimalnoe impulsnoe upravlenie s prilozheniyami, Fizmatlit, M., 2003 | MR | Zbl

[31] Silva G.N., Vinter R.B., “Measure driven differential inclusions”, J. Math. Anal. and Appl., 202 (1996), 727–746 | DOI | MR | Zbl

[32] Pereira F.L., Silva G.N., “Necessary conditions of optimality for vector-valued impulsive control problems”, Syst. and Control Lett., 40 (2000), 205–215 | DOI | MR | Zbl

[33] Motta M., Rampazzo F., “Space–time trajectories of nonlinear systems driven by ordinary and impulsive controls”, Diff. and Integral Equat., 8 (1995), 269–288 | MR | Zbl

[34] Motta M., Rampazzo F., “Dynamic programming for nonlinear systems driven by ordinary and impulsive controls”, SIAM J. Control and Optim., 34 (1996), 199–225 | DOI | MR | Zbl

[35] Motta M., Sartori C., “Discontinuous solutions to unbounded differential inclusions under state constraints. Applications to optimal control problems”, Set-Valued Anal., 7 (1999), 295–322 | DOI | MR | Zbl

[36] Stefanova A.V., “Uravnenie Gamiltona–Yakobi–Bellmana v nelineinoi zadache impulsnogo upravleniya”, Tr. In-ta matematiki i mekhaniki UrO RAN, 5, 1998, 301–318 | Zbl

[37] Kurzhanski A.B., Daryin A.N., “Dynamic programming for impulse controls”, Annu. Rev. Control., 32:2 (2008), 213–227 | DOI

[38] Antipina N.V., Dykhta V.A., “Dostatochnye usloviya optimalnosti dlya zadach impulsnogo upravleniya”, Izv. RAN. Teoriya i sist. upr., 2004, no. 4, 76–83 | MR

[39] Dykhta V.A., Samsonyuk O.N., “Otsenki mnozhestv dostizhimosti i dostatochnye usloviya optimalnosti dlya nelineinykh upravlyaemykh sistem s razryvnymi traektoriyami”, Vestn. Tambov. un-ta. Est. i tekhn. nauki., 14:4 (2009), 707–709

[40] Pereira F.L., Silva Zh.N., “Ustoichivost po Lyapunovu impulsnykh sistem, upravlyaemykh meroi”, Dif. uravneniya, 40:8 (2004), 1059–1067 | MR | Zbl

[41] Pereira F.L., Silva Zh.N., Oliveira V., “Invariantnost dlya impulsnykh upravlyaemykh sistem”, Avtomatika i telemekhanika, 2008, no. 5, 57–71 | Zbl

[42] Dykhta V.A., Samsonyuk O.N., “Some applications of Hamilton–Jacobi inequalities for classical and impulsive optimal control problems”, Eur. J. Control., To appear

[43] Miller B.M., “The generalized solutions of nonlinear optimization problems with impulse control”, SIAM J. Control and Optim., 34 (1996), 1420–1440 | DOI | MR | Zbl

[44] Pereira F.L., de Matos A.C., Silva G.N., “Hamilton–Jacobi conditions for an impulsive control problem”, Nonlinear control systems 2001 (NOLCOS 2001), Proc. 5th IFAC Symp., v. 3, Pergamon Press, Oxford, 2002, 1297–1302

[45] Arutyunov A., Karamzin D., Pereira F., “A nondegenerate maximum principle for the impulse control problem with state constraints”, SIAM J. Control and Optim., 43 (2005), 1812–1843 | DOI | MR | Zbl