Compound Lyapunov type functions in control problems of impulsive dynamical systems

O. N. Samsonyuk

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 5, pp. 170-178

Résumé

The paper is devoted to the study of impulsive dynamical systems with trajectories of bounded variation and impulsive controls (regular vector measures). In the paper, strongly monotone Lyapunov type functions relative to impulsive control system are introduced and infinitesimal conditions of the strong monotonicity of Lyapunov type functions are presented. We focus mainly on the study of compound (defined piecewise in the variable t) Lyapunov type functions, which are more applicable for dynamical systems with discontinuous trajectories, and give their applications to outer estimations of the reachable sets.

Détail

Keywords: trajectory of bounded variation, Hamilton–Jacobi inequalities, strongly monotone Lyapunov type functions.

Bibliographie

[1] Clarke F. H., Ledyaev Yu. S., Stern R. J., Wolenski P. R., Nonsmooth Analysis and Control Theory, Grad. Texts in Math., 178, Springer-Verlag, N.Y., 1998, 276 pp. | MR | Zbl

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

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

[4] Vinter R. B., Optimal Control, Birkhuser, Boston, 2000 | MR | Zbl

[5] Milyutin A. A., Osmolovskii N. P., Calculus of Variations and Optimal Control, American Mathematical Society, Providence, Rhode Island, 1998 | MR | Zbl

[6] 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

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

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

[9] Pereira F. L., Matos A. C., Silva G. N., “Hamilton–Jacobi conditions for an impulsive control problem”, Nonlinear Control Systems, Fevereiro, 2002, 1297–1302

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

[11] Daryin A. N., Kurzhanski A. B., Seleznev A. V., “A dynamic programming approach to the impulse control synthesis problem”, Proc. Joint 44th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC05 (Seville, December 12–15, 2005), 8215–8220

[12] Daryin A. N., Kurzhanski A. B., “Dynamic programming for impulse control”, Ann. Reviews in Control, 32:2 (2008), 213–227 | DOI

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

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

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

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

[17] Samsonyuk O. N., “A Hamilton–Jacobi inequality and sufficient optimality conditions for impulsive processes”, Discrete and Continuous Dynamical Systems. Series A, Special Issue: Proc. Workshop on Control, Nonsmooth Analysis and Optimization (May 4–8, 2009, Porto, Portugal) (to appear)

[18] Dykhta V. A., Samsonyuk O. N., “Neravenstva Gamiltona–Yakobi v zadachakh upravleniya impulsnymi dinamicheskimi sistemami”, Tr. Matemat. in-ta im. V. A. Steklova RAN, 2010 (to appear)

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

[20] Sesekin A. N., “Dinamicheskie sistemy s nelineinoi impulsnoi strukturoi”, Tr. In-ta matematiki i mekhaniki UrO RAN, 6, Ekaterinburg, 2000, 497–510 | MR | Zbl

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

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

[23] Arutyunov A., Karamzin D. Yu., Pereira F. L., “On a Generalization of the Impulsive Control Concept: Controlling System Jumps”, Discrete and Continuous Dynamical Systems. Series A, Special Issue: Proc. Workshop on Control, Nonsmooth Analysis and Optimization (May 4–8, 2009, Porto, Portugal) (to appear)