On dynamical regularization under random noise

A. V. Kryazhimskii ; Yu. S. Osipov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and topology. II, Tome 271 (2010), pp. 134-147.

Résumé

We consider the problem of constructing a robust dynamic approximation of a time-varying input to a control system from the results of inaccurate observation of the states of the system. In contrast to the earlier studied cases in which the observation errors are assumed to be small in the metric sense, the errors in the present case are allowed to take, generally, large values and are subject to a certain probability distribution. The observation errors occurring at different instants are supposed to be statistically independent. Under the assumption that the expected values of the observation errors are small, we construct a dynamical algorithm for approximating the normal (minimal in the sense of the mean-square norm) input; the algorithm ensures an arbitrarily high level of the mean-square approximation accuracy with an arbitrarily high probability.

Bibliographie

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

[2] Tikhonov A.N., Arsenin V.Ya., Metody resheniya nekorrektnykh zadach, Nauka, M., 1974 | MR

[3] Kotelnikova A.N., “Odna zadacha ob igrovoi stokhasticheskoi stabilizatsii upravlyaemogo ob'ekta”, Trudy In-ta matematiki i mekhaniki UrO RAN, 10, no. 2, 2004, 69–82 | Zbl

[4] Krasovskii N.N., Kotelnikova A.N., “Ob odnoi zadache ob ustoichivom otslezhivanii dvizheniya”, Trudy In-ta matematiki i mekhaniki UrO RAN, 12, no. 1, 2006, 142–157

[5] Nevë Zh., Matematicheskie osnovy teorii veroyatnostei, Mir, M., 1969 | MR | Zbl

[6] Liptser R.Sh., Shiryaev A.N., Statistika sluchainykh protsessov: Nelineinaya filtratsiya i smezhnye voprosy, Nauka, M., 1974 | MR | Zbl

[7] Krasovskii N.N., Subbotin A.I., Rossokhin V.F., “Stokhasticheskie strategii v differentsialnykh igrakh”, DAN SSSR, 220:5 (1975), 1023–1026 | MR

[8] Krasovskii N.N., Subbotin A.I., “Differentsialnye igry v smeshannykh strategiyakh”, Problemy analiticheskoi mekhaniki, teorii ustoichivosti i upravleniya, Nauka, M., 1975, 11–18 | MR

[9] Krasovskii N.N., “Smeshannye strategii v differentsialnykh igrakh”, DAN SSSR, 235:3 (1977), 519–522 | MR

[10] Kryazhimskii A.V., “O stokhasticheskoi approksimatsii v differentsialnykh igrakh”, DAN SSSR, 241:5 (1978), 1020–1023 | MR

[11] Kryazhimskii A.V., Osipov Yu.S., “O modelirovanii upravleniya v dinamicheskoi sisteme”, Izv. AN SSSR. Ser. tekhn. kibern., 1983, no. 2, 51–60 | MR

[12] Osipov Yu.S., Kryazhimskii A.V., Inverse problems for ordinary differential equations: Dynamical solutions, Gordon and Breach, London, 1995 | MR | Zbl

[13] Osipov Yu.S., Vasilev F.P., Potapov M.M., Osnovy metoda dinamicheskoi regulyarizatsii, Izd-vo Mosk. un-ta, M., 1999

[14] Maksimov V.I., Zadachi dinamicheskogo vosstanovleniya vkhodov beskonechnomernykh sistem, In-t matematiki i mekhaniki UrO RAN, Ekaterinburg, 2000

[15] Osipov Yu.S., Kryazhimskii A.V., Maksimov V.I., “Dinamicheskie obratnye zadachi dlya parabolicheskikh sistem”, Dif. uravneniya, 36:5 (2000), 579–597 | MR | Zbl

[16] Kryazhimskiy A., Maksimov V., “On rough inversion of a dynamical system with a disturbance”, J. Inverse and Ill-Posed Probl., 16 (2008), 587–600 | DOI | MR | Zbl