On elimination of state constraints in the construction of reachable sets

M. I. Gusev

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 4, pp. 106-115

Résumé

The paper is devoted to the problem of approximating reachable sets of a nonlinear control system with state constraints given as a solution set of a nonlinear inequality. A state constraint elimination procedure based on the introduction of an auxiliary constraint-free control system is proposed. The equations of the auxiliary system depend on a small parameter. It is shown that the reachable set of the original system can be approximated in the Hausdorff metric by reachable sets of the auxiliary control system as the small parameter tends to zero. Estimates of the convergence rate are given.

Détail

Keywords: reachable set, state constraints, penalty function, approximation, Hausdorff metric.

Bibliographie

[1] Krasovskii N. N., Igrovye zadachi o vstreche dvizhenii, Nauka, M., 1970, 420 pp. | MR | Zbl

[2] Kurzhanskii A. B., Upravlenie i nablyudenie v usloviyakh neopredelennosti, Nauka, M., 1977, 392 pp. | MR | Zbl

[3] Chernousko F. L., Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem, Nauka, M., 1988, 319 pp. | MR | Zbl

[4] Krasovski N. N., Subbotin A. I., Game-theoretical control problems, Springer-Verlag, New York, 1988, 517 pp. | MR | Zbl

[5] Kurzhanski A. B., Valyi I., Ellipsoidal calculus for estimation and control, Birkhäuser, Boston, 1997, 321 pp. | MR | Zbl

[6] Lotov A. V., “Chislennyi metod postroeniya mnozhestv dostizhimosti dlya lineinykh upravlyaemykh sistem s fazovymi ogranicheniyami”, Zhurn. vychisl. matematiki i mat. fiziki, 15:1 (1975), 67–78 | MR | Zbl

[7] Matviichuk A. R., Ushakov V. N., “O postroenii razreshayuschikh upravlenii v zadachakh upravleniya s fazovymi ogranicheniyami”, Izv. RAN. Teoriya i sistemy upravleniya, 2006, no. 1, 5–20 | MR

[8] Kurzhanski A. B., Mitchell I. M., Varaiya P., “Optimization techniques for state-constrained control and obstacle problems”, J. Optim. Theory Appl., 128:3 (2006), 499–521 | DOI | MR | Zbl

[9] Baier R., Chahma I. A., Lempio F., “Stability and convergence of Euler's method for state-constrained differential inclusions”, SIAM J. Optim., 18:3 (2007), 1004–1026 | DOI | MR | Zbl

[10] Bonneuil N., “Computing reachable sets as capture-viability kernels in reverse time”, Appl. Math., 3:11 (2012), 1593–1597 | DOI

[11] Kostousova E. K., “Vneshnee i vnutrennee otsenivanie oblastei dostizhimosti pri pomoschi parallelotopov”, Vychisl. tekhnologii, 3:2 (1998), 11–20 | MR | Zbl

[12] Gusev M. I., “Vneshnie otsenki mnozhestv dostizhimosti nelineinykh upravlyaemykh sistem”, Avtomatika i telemekhanika, 2012, no. 3, 39–51

[13] Kurzhanskii A. B., Filippova T. F., “Ob opisanii mnozhestva vyzhivayuschikh traektorii differentsialnogo vklyucheniya”, Dokl. AN SSSR, 289:1 (1986), 38–41 | MR

[14] Kurzhanskii A. B., Filippova T. F., “Ob opisanii puchka vyzhivayuschikh traektorii upravlyaemoi sistemy”, Differents. uravneniya, 23:8 (1987), 1303–1315 | MR

[15] Gusev M. I., “O metode shtrafnykh funktsii v zadache postroeniya mnozhestv dostizhimosti upravlyaemykh sistem s fazovymi ogranicheniyami”, Tr. In-ta matematiki i mekhaniki UrO RAN, 19, no. 1, 2013, 81–86

[16] Gusev M. I., “Vnutrennie approksimatsii mnozhestv dostizhimosti upravlyaemykh sistem s fazovymi ogranicheniyami”, Tr. In-ta matematiki i mekhaniki UrO RAN, 19, no. 4, 2013, 73–88

[17] Forcellini F., Rampazzo F., “On nonconvex differential inclusions whose state is constrained in the closure of an open set”, Differential Integral Equations, 12:4 (1999), 471–497 | MR | Zbl

[18] Frankowska H., Vinter R. B., “Existence of neighboring feasible trajectories: applications to dynamic programming for state-constrained optimal control problems”, J. Optim. Theory Appl., 104:1 (2000), 21–40 | DOI | MR | Zbl

[19] Bettiol P., Bressan A., Vinter R., “Trajectories satisfying a state constraint: $W^{(1,1)}$ estimates and counterexamples”, SIAM J. Control Optim., 48:7 (2010), 4664–4679 | DOI | MR | Zbl

[20] Stern R. J., “Characterization of the state constrained minimal time function”, SIAM J. Control Optim., 43:2 (2004), 697–707 | DOI | MR | Zbl

[21] Ioffe A. D., Tikhomirov V. M., Teoriya ekstremalnykh zadach, Nauka, M., 1974, 480 pp. | MR | Zbl

[22] Sontag E. D., “A “universal” construction of Artstein's theorem on nonlinear stabilization”, System and Control Letters, 13:2 (1989), 117–123 | DOI | MR | Zbl

[23] Li E. B., Markus L., Osnovy teorii optimalnogo upravleniya, Nauka, M., 1972, 576 pp. | MR

[24] Gusev M.I., “O zadache dostizhimosti pri fazovykh ogranicheniyakh”, XII Vseros. soveschanie po problemam upravleniya – VSPU-2014, Sb. dokl. (Moskva, 16–19 iyunya, 2014 g.), [Elektron. resurs], 610–621