Weak invariance of a cylindrical set with smooth boundary with respect to a control system

A. A. Uspenskii

Trudy Instituta matematiki i mehaniki, Tome 23 (2017) no. 1, pp. 241-250

Résumé

We consider the problem of constructing resolving sets for a differential game or an optimal control problem based on information on the dynamics of the system, control resources, and boundary conditions. The construction of largest possible sets with such properties (the maximal stable bridge in a differential game or the controllability set in a control problem) is a nontrivial problem due to their complicated geometry; in particular, the boundaries may be nonconvex and nonsmooth. In practical engineering tasks, which permit some tolerance and deviations, it is often admissible to construct a resolving set that is not maximal. The constructed set may possess certain characteristics that would make the formation of control actions easier. For example, the set may have convex sections or a smooth boundary. In this context, we study the property of stability (weak invariance) for one class of sets in the space of positions of a differential game. Using the notion of stability defect of a set introduced by V.N. Ushakov, we derive a criterion of weak invariance with respect to a conflict-controlled dynamic system for cylindrical sets. In a particular case of a linear control system, we obtain easily verified sufficient conditions of weak invariance for cylindrical sets with ellipsoidal sections. The proof of the conditions is based on constructions and facts of subdifferential calculation. An illustrating example is given.

Détail

Keywords: stable set, weak invariance, differential game, Hamiltonian, stability defect, cylindrical set, ellipsoid, subdifferential.

Bibliographie

[1] Krasovskii N.N., Subbotin A.I., Pozitsionnye differentsialnye igry, Nauka, M., 1974, 456 pp. | MR

[2] Krasovskii N.N., “Igrovye zadachi dinamiki. I”, Izv. AN SSSR. Tekhn. kibernetika, 1969, no. 5, 3–12

[3] Krasovskii N.N., Subbotin A.I., “O strukture differentsialnykh igr”, Dokl. AN SSSR, 190:3 (1970), 523–526

[4] Krasovskii N.N., Subbotin A.I., “Approksimatsiya v differentsialnykh igrakh”, Prikl. matematika i mekhanika, 37:2 (1973), 197–204

[5] Krasovskii N.N., “K zadache unifikatsii differentsialnykh igr”, Dokl. AN SSSR, 226:6 (1976), 1260–1263 | MR

[6] Subbotin A.I., Obobschennye resheniya uravnenii v chastnykh proizvodnykh pervogo poryadka. Perspektivy dinamicheskoi optimizatsii, In-t kompyut. issled., M.; Izhevsk, 2003, 336 pp.

[7] Tarasev A.M., Uspenskii A.A., Ushakov V.N., “Approksimatsionnye operatory i konechno-raznostnye skhemy dlya postroeniya obobschennykh reshenii uravnenii Gamiltona - Yakobi”, Izv. RAN Tekhnicheskaya kibernetika, 1994, no. 3, 173–185 | Zbl

[8] Guseinov H.G., Subbotin A.I., Ushakov V.N., “Derivatives for multivalued mappings with applications to game-theoretical problems of control”, Problem Control Inform. Theory, 14:6 (1985), 405–419 | MR

[9] Roxin E., “A uniqueness theorem for differential inclusions”, Diff. Equ., 1:2 (1965), 115–150 | DOI | MR | Zbl

[10] Aubin J.-P., Cellina A., Differential inclusions. Set valued maps and viability theory, Berlin, 1984, 342 pp. | MR

[11] Nguen Chan, “Invariantnye i ustoichivye semeistva mnozhestv otnositelno differentsialnykh vklyuchenii”, Differents. uravneniya, 19:8 (1983), 1357–1366 | MR | Zbl

[12] Kurzhanskii A.B., Filippova T.F., “Differentsialnye vklyucheniya s fazovymi ogranicheniyami. Metod vozmuschenii”, Tr. MIAN: Optimalnoe upravlenie i differents. uravneniya, 211 (1995), 304–315 | MR

[13] Frankowska H., Plaskacz S., Rzezuchowski T., “Theoremes de viabilite mesurables et l'equation d'Hamilton-Jacobi-Bellman”, C. r. Acad. sei. Paris. Ser. 1, 315 (1992), 131–134 | MR | Zbl

[14] Kurzhanski A., Valyi I., Ellipsoidal calculus for estimation and control, Birkhauser, Basel, 1997, 321 pp. | MR | Zbl

[15] Chernousko F.L., Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem: Metod ellipsoidov, Nauka, M., 1988, 320 pp.

[16] Polyak B.T., Khlebnikov M.V., Scherbakov P.S., Upravlenie lineinymi sistemami pri vneshnikh vozmuscheniyakh: Tekhnika lineinykh matrichnykh neravenstv, LENAND, M., 2014, 560 pp.

[17] Ushakov V.N., “K zadache postroeniya stabilnykh mostov v differentsialnoi igre sblizheniya-ukloneniya”, Izv. AN SSSR. Tekhn. kibernetika, 1980, no. 4, 29–36 | Zbl

[18] Ushakov V.N., Latushkin Ya.A., “Defekt stabilnosti mnozhestv v igrovykh zadachakh upravleniya”, Tr. In-ta matematiki i mekhaniki, 12:2 (2006), 178–194 | Zbl

[19] Ushakov V.N., Uspenskii A.A., “K svoistvu stabilnosti v differentsialnykh igrakh”, Dokl. AN, 443:5 (2012), 549–554 | Zbl

[20] Strekalovskii A.S., Elementy nevypukloi optimizatsii, Nauka, Novosibirsk, 2003, 356 pp.

[21] Demyanov V.F., Vasilev L.V., Nedifferentsiruemaya optimizatsiya, Nauka, M., 1981, 384 pp.