Invariant and statistically weakly invariant sets of control systems

L. I. Rodina

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 40 (2012) no. 2, pp. 3-164.

Résumé

The paper deals with the expansion of the concept of the set invariance with respect to control systems and differential inclusions. Said expansions that statistically invariant and statistically weakly invariant sets are studied. The sufficient conditions for existence of invariant (in the specified sense) sets, which are formulated in terms of the Hausdorff–Bebutov metric, Lyapunov functions and the Clarke derivative, of the given functions are obtained. The work covers both determined systems and the systems with random parameters, for which the concept of statistical invariance with probability one is investigated. The problems about complete controllability of time-varying linear system and about the existence of non-predicting control for linear system with random parameters are considered, too.

Détail

Keywords: control systems, dynamical systems, differential inclusions, statistically invariant sets.

Bibliographie

[1] Agrachev A.F., Sachkov A.L., Geometricheskaya teoriya upravleniya, Fizmatlit, M., 2005, 391 pp.

[2] Adomian Dzh., Stokhasticheskie sistemy, Mir, M., 1987, 376 pp. | MR

[3] Andreev N.I., Teoriya statisticheski optimalnykh sistem upravleniya, Nauka, M., 1980, 415 pp. | MR

[4] Anosov D.V., Aranson S.Kh., Bronshtein I.U., Grines V.Z., Dinamicheskie sistemy–1, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 1, VINITI, M., 1985, 244 pp. | MR

[5] Arnold V.I., Avets A., Ergodicheskie problemy klassicheskoi mekhaniki, Regulyarnaya i khaoticheskaya dinamika, Izhevsk, 1999, 284 pp. | Zbl

[6] Asnis I.A., Dmitruk A.V., Osmolovskii N.P., “Reshenie s pomoschyu printsipa maksimuma zadachi ob energeticheski optimalnom upravlenii dvizheniem poezda”, Zhurnal vychislitelnoi matematiki i matematicheskoi fiziki, 25:11 (1985), 1644–1655 | MR | Zbl

[7] Astapov Yu.M., Medvedev V.S., Statisticheskaya teoriya sistem avtomaticheskogo regulirovaniya i upravleniya, Nauka, M., 1982, 304 pp. | MR

[8] Baranova O.V., “O ravnomernoi globalnoi upravlyaemosti lineinoi sistemy so statsionarnymi sluchainymi parametrami”, Differentsialnye uravneniya, 27:11 (1991), 1843–1850 | MR | Zbl

[9] Barbashin E.A., Funktsii Lyapunova, Nauka, M., 1970, 240 pp. | MR | Zbl

[10] Bebutov M.V., “O dinamicheskikh sistemakh v prostranstve nepreryvnykh funktsii”, Byulleten Mekhaniko-matematicheskogo fakulteta MGU, 5 (1941), 1–52

[11] Billingslei P., Ergodicheskaya teoriya i informatsiya, Mir, M., 1969, 238 pp. | MR

[12] Birkgof D., Dinamicheskie sistemy, Izdatelskii dom «Udmurtskii universitet», Izhevsk, 1999, 408 pp. | Zbl

[13] Blagodatskikh V.I., Filippov A.F., “Differentsialnye vklyucheniya i optimalnoe upravlenie”, Trudy Matematicheskogo instituta im. V. A. Steklova AN SSSR, 169, 1985, 194–252 | MR | Zbl

[14] Borisovich Yu.G., Gelman B.D., Myshkis A.D., Obukhovskii V.V., “Mnogoznachnye otobrazheniya”, Itogi nauki i tekhniki. Matematicheskii analiz, 19, VINITI, M., 1982, 127–230 | MR

[15] Borisovich Yu.G., Gelman B.D., Myshkis A.D., Obukhovskii V.V., Vvedenie v teoriyu mnogoznachnykh otobrazhenii i differentsialnykh vklyuchenii, Knizhnyi dom «LIBROKOM», M., 2011, 224 pp. | MR | Zbl

[16] Borukhov V.T., “K voprosu o neobkhodimykh usloviyakh upravlyaemosti dlya lineinykh nestatsionarnykh dinamicheskikh sistem”, Vestsi AN BSSR. Ser. fiz.-mat. navuk, 1979, no. 6, 27–30 | MR

[17] Burago D.Yu., Burago Yu.D., Ivanov S.V, Kurs metricheskoi geometrii, In-t kompyuternykh issledovanii, M. – Izhevsk, 2004, 496 pp.

[18] Valeev K.G., Karelova O.L., Gorelov V.I., Optimizatsiya lineinykh sistem so sluchainymi koeffitsientami, Izd-vo RUDN, M., 1996, 231 pp.

[19] Varga Dzh., Optimalnoe upravlenie differentsialnymi i funktsionalnymi uravneniyami, Nauka, M., 1977, 623 pp. | MR

[20] Venttsel A.D., Kurs teorii sluchainykh protsessov, Nauka, M., 1975, 320 pp. | MR

[21] Vershik A.M., Kornfeld I.P., Sinai Ya.G., “Obschaya ergodicheskaya teoriya grupp preobrazovanii s invariantnoi meroi”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 2, VINITI, M., 1985, 5–111 | MR

[22] A. M. Vershik, S. A. Yuzvinskii, “Dinamicheskie sistemy s invariantnoi meroi”, Itogi nauki. Ser. Matematika. Mat. anal. 1967, VINITI, M., 1969, 133–187 | MR | Zbl

[23] Gabasov R., “K teorii upravlyaemosti dinamicheskikh sistem”, Differentsialnye uravneniya, 4:9 (1968), 1499–1507

[24] Gabasov R., Kirillova F.M., Kachestvennaya teoriya optimalnykh protsessov, Nauka, M., 1971, 508 pp. | MR

[25] Gabasov R., Kirillova F.M., Osobye optimalnye upravleniya, Nauka, M., 1973, 256 pp. | MR

[26] Gaishun I.V., Vvedenie v teoriyu lineinykh nestatsionarnykh sistem, Institut matematiki NAN Belarusi, Minsk, 1999, 408 pp. | MR

[27] Galperin E.A., Krasovskii N.N., “O stabilizatsii statsionarnykh dvizhenii v nelineinykh upravlyaemykh sistemakh”, Prikladnaya matematika i mekhanika, 27 (1963), 1–24

[28] Gantmakher F.R., Teoriya matrits, Nauka, M., 1966, 576 pp. | MR

[29] Gelman B.D., “Ob odnom klasse mnogoznachnykh otobrazhenii s nekompaktnymi obrazami”, Vestnik VGU, seriya: fizika, matematika, 2008, no. 1, 162–169

[30] Gelman B.D., Obukhovskii V.V., “O novykh rezultatakh v teorii mnogoznachnykh otobrazhenii. II”, Analiz i prilozheniya. Itogi nauki i tekhniki. Matematicheskii analiz, 29, VINITI, M., 1991, 107–159 | MR

[31] Gikhman I.I., Skorokhod A.V., Upravlenie sluchainymi protsessami, Naukova dumka, Kiev, 1997, 252 pp.

[32] Glyzin S.D., Kolesov A.Yu., Rozov N.Kh., “K voprosu o realizuemosti stsenariya razvitiya turbulentnosti po Landau”, Teoreticheskaya i matematicheskaya fizika, 158:2 (2009), 292–311 | MR | Zbl

[33] Gurman V.I., Sachkov Yu.L., “Predstavlenie i realizatsiya obobschennykh reshenii upravlyaemykh sistem s neogranichennym godografom”, Avtomatika i telemekhanika, 2008, no. 4, 72–80 | MR | Zbl

[34] Gurman V.I., Trushkova E.A., “Otsenki mnozhestv dostizhimosti upravlyaemykh sistem”, Differentsialnye uravneniya, 45:11 (2009), 1601–1609 | MR | Zbl

[35] Gusev M.I., “Otsenki pogreshnosti dlya mnozhestv dostizhimosti upravlyaemykh sistem s fazovymi ogranicheniyami”, Trudy Instituta matematiki i mekhaniki UrO RAN, 12:2 (2006), 64–77

[36] Gusev M.I., “O vneshnikh otsenkakh mnozhestv dostizhimosti nelineinykh upravlyaemykh sistem”, Trudy Instituta matematiki i mekhaniki UrO RAN, 17:1 (2011), 60–69 | MR

[37] Guseinov Kh.G., Moiseev A.N., Ushakov V.N., “Ob approksimatsii oblastei dostizhimosti sistem upravleniya”, Prikladnaya matematika i mekhanika, 1998, no. 2, 179–186 | MR

[38] Guseinov Kh.G., Nigal Ege., “O svoistvakh pozitsionno slabo invariantnykh mnozhestv otnositelno upravlyaemykh sistem, opisyvaemykh differentsialnymi vklyucheniyami”, Differentsialnye uravneniya, 43:3 (2007), 291–302 | MR | Zbl

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

[40] Guselnikov N.S., “Treugolnye funktsii mnozhestva i teoremy Nikodima, Bruksa-Dzhevetta i Vitali-Khana-Saksa o skhodyaschikhsya posledovatelnostyakh mer”, Matematicheskii sbornik, 202:6 (2011), 29–50 | MR

[41] Davydov A.A., “Osobennosti granitsy dostizhimosti v dvumernykh upravlyaemykh sistemakh”, Uspekhi matematicheskikh nauk, 37:3(225) (1982), 183–184 | MR | Zbl

[42] Davydov A.A., Pasters R., Petrenko I.A., “Optimalnoe raspredelenie vybrosa zagryazneniya v odnomernyi potok”, Trudy Instituta matematiki i mekhaniki UrO RAN, 16:5 (2010), 30–35

[43] Danford N., Shvarts Dzh.T., Lineinye operatory. Obschaya teoriya, Editorial URSS, M., 2004, 896 pp.

[44] Demyanov V.F., Rubinov A.M., Osnovy negladkogo analiza i kvazidifferentsialnoe ischislenie, Nauka, M., 1990, 432 pp. | MR

[45] Demidovich B.P., Lektsii po matematicheskoi teorii ustoichivosti, Nauka, M., 1967, 472 pp. | MR

[46] Dimentberg M.F., Sluchainye protsessy v dinamike sistem s peremennymi parametrami, Nauka, M., 1989, 176 pp. | MR | Zbl

[47] Dmitruk A.V., “Printsip maksimuma dlya obschei zadachi optimalnogo upravleniya s fazovymi i regulyarnymi smeshannymi ogranicheniyami”, «Optimalnost upravlyaemykh dinamicheskikh sistem», no. 14, VNIISI, M., 1990, 26–42

[48] Evlanov L.G., Konstantinov V.M., Sistemy so sluchainymi parametrami, Nauka, M., 1976, 568 pp. | MR | Zbl

[49] Zhikov V.V., “K probleme pochti-periodichnosti dlya differentsialnykh i operatornykh uravnenii”, Sbornik nauchnykh trudov VPI, 8 (1969), 94–188

[50] Zhikov V.V., Pyatnitskii A.L., “Usrednenie sluchainykh singulyarnykh struktur i sluchainykh mer”, Izvestiya Rossiiskoi akademii nauk. Seriya matematicheskaya, 70:1 (2006), 23–74 | MR | Zbl

[51] Zabello L.E., “Ob upravlyaemosti lineinykh nestatsionarnykh sistem”, Avtomatika i telemekhanika, 1973, no. 8, 13–19 | MR | Zbl

[52] Zabello L.E., “K teorii upravlyaemosti nestatsionarnykh sistem”, Doklady AN BSSR, 24:6 (1980), 497–499 | MR | Zbl

[53] Zavyalova T.V., Kats I.Ya., Timofeeva G.A., “Ob ustoichivosti dvizheniya stokhasticheskoi sistemy so sluchainym usloviem skachka fazovoi traektorii”, Avtomatika i telemekhanika, 2002, no. 7, 33–46 | MR

[54] Ivanov A.G., “Dinamicheskaya sistema sdvigov i suschestvovanie resheniya zadachi pochti periodicheskoi optimizatsii”, Izvestiya vysshikh uchebnykh zavedenii. Matematika, 2005, no. 10(521), 29–46

[55] Ivanov A.G., Tonkov E.L., “Metricheskie svoistva lineinykh upravlyaemykh sistem”, Uspekhi matematicheskikh nauk, 46:6(282) (1991), 133–204

[56] Ivanov A.G., Tonkov E.L., “O mnozhestve upravlyaemosti lineinoi pochti periodicheskoi sistemy”, Differentsialnye uravneniya, 27:10 (1991), 1692–1699 | MR

[57] Ivanov A.G., Tonkov E.L., “O ravnomernoi lokalnoi upravlyaemosti lineinoi sistemy”, Differentsialnye uravneniya, 28:9 (1992), 1499–1507 | MR | Zbl

[58] Ivanov A.G., Tonkov E.L., Shneiberg I.Ya., “O mere mnozhestva globalno upravlyaemykh sistem”, Nelineinye kolebaniya i teoriya upravleniya. Izhevsk, 1981, no. 3, 3–32 | MR

[59] Kazakov I.E., Statisticheskaya teoriya sistem upravleniya v prostranstve sostoyanii, Nauka, M., 1975, 432 pp. | MR | Zbl

[60] Kazakov I.E., Dostupov B.G., Statisticheskaya dinamika nelineinykh avtomaticheskikh sistem, Fizmatgiz, M., 1962, 332 pp.

[61] Kalman R.E., “Ob obschei teorii sistem upravleniya”, Trudy I Mezhdunarodnogo kongressa IFAK, v. 2, Izdatelstvo AN SSSR, 1961, 521–547

[62] Kalman R., Falb P., Arbib M., Ocherki po matematicheskoi teorii sistem, Mir, M., 1971, 400 pp. | MR | Zbl

[63] Kamenskii M.I., Obukhovskii V.V., “Ob operatore sdviga po traektoriyam upravlyaemykh sistem”, Differentsialnye uravneniya, 32:6 (1996), 747–754 | MR

[64] Katok A.B., Sinai Ya.G., Stepin A.M., “Teoriya dinamicheskikh sistem i obschikh grupp preobrazovanii s invariantnoi meroi”, Itogi nauki i tekhniki. Matematicheskii analiz, 13, VINITI, M., 1975, 129–262 | MR

[65] Katok A.B., Khasselblat B., Vvedenie v sovremennuyu teoriyu dinamicheskikh sistem, Faktorial, M., 1999, 767 pp.

[66] Kats I.Ya., Metod funktsii Lyapunova v zadachakh ustoichivosti i stabilizatsii sistem sluchainoi struktury, Izd-vo Uralskoi gos. akademii putei soobscheniya, Ekaterinburg, 1998, 222 pp.

[67] Kats I.Ya., Krasovskii N.N., “Pro ustoichivost sistem so sluchainymi parametrami”, Prikladnaya matematika i mekhanika, 1960, no. 5, 809–823 | Zbl

[68] Klark F., Optimizatsiya i negladkii analiz, Nauka, M., 1988, 300 pp. | MR | Zbl

[69] Kolmogorov A.N., “Ob entropii na edinitsu vremeni kak metricheskom invariante avtomorfizmov”, Doklady AN SSSR, 124:4 (1959), 754–755 | MR | Zbl

[70] Kornev S.V., Obukhovskii V.V., “O lokalizatsii metoda napravlyayuschikh funktsii v zadache o periodicheskikh resheniyakh differentsialnykh vklyuchenii”, Izvestiya vysshikh uchebnykh zavedenii. Matematika, 2009, no. 5, 23–32 | MR | Zbl

[71] Kornfeld I.P., “Ob invariantnykh merakh minimalnykh dinamicheskikh sistem”, Doklady AN SSSR, 202:2 (1972), 280–283 | MR

[72] Kornfeld I.P., Sinai Ya.G., Fomin S.V., Ergodicheskaya teoriya, Nauka, M., 1980, 384 pp. | MR

[73] Korolyuk V.S., Portenko N.I., Skorokhod A.V., Turbin A.F., Spravochnik po teorii veroyatnostei i matematicheskoi statistike, Nauka, M., 1985, 640 pp. | MR | Zbl

[74] Krasovskii A.A., Statisticheskaya teoriya perekhodnykh protsessov v sistemakh upravleniya, Nauka, M., 1968, 240 pp. | MR

[75] Krasovskii A.A., Fazovoe prostranstvo i statisticheskaya teoriya dinamicheskikh sistem, Nauka, M., 1974, 232 pp. | Zbl

[76] Krasovskii A.N., Krasovskii N.N., Tretyakov V.E., “Stokhasticheskii programmnyi sintez dlya determinirovannoi pozitsionnoi differentsialnoi igry”, Prikladnaya matematika i mekhanika, 45:4 (1981), 579–586 | MR

[77] Krasovskii N.N., Teoriya upravleniya dvizheniem, Nauka, M., 1968, 476 pp. | MR

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

[79] Krasovskii N.N., “Stokhasticheskii programmnyi sintez dlya determinirovannoi pozitsionnoi differentsialnoi igry”, Prikladnaya matematika i mekhanika, 46:6 (1982), 885–892 | MR

[80] Krasovskii N.N., Upravlenie dinamicheskoi sistemoi, Nauka, M., 1985, 520 pp. | MR

[81] Krasovskii N.N., Lidskii E.A., “Analiticheskoe konstruirovanie regulyatorov v stokhasticheskikh sistemakh pri ogranicheniyakh na skorost izmeneniya upravlyayuschego vozdeistviya”, Prikladnaya matematika i mekhanika, 25:3 (1961), 420–432 | MR

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

[83] Kultyshev S.Yu., Tonkov E.L., “Upravlyaemost lineinoi nestatsionarnoi sistemy”, Differentsialnye uravneniya, 11:7 (1975), 1210–1216

[84] Kurzhanskii A.B., Upravlenie i nablyudenie v usloviyakh neopredelennosti, Nauka, M., 1979, 392 pp. | MR

[85] Kurzhanskii A.B., Varaiya P., “O probleme dostizhimosti pri postoyanno deistvuyuschikh vozmuscheniyakh”, Doklady RAN, 372:4 (2000), 446–450 | MR

[86] Kurzhanskii A.B., Varaiya P., “Zadachi dinamiki i upravleniya v gibridnykh sistemakh”, Trudy mezhdunarodnogo seminara «Teoriya upravleniya i teoriya obobschennykh reshenii uravnenii Gamiltona–Yakobi», Izd-vo Uralskogo universiteta, Ekaterinburg, 2005, 26–33

[87] Kurzhanskii A.B., Tochilin P.A., “Slabo invariantnye mnozhestva gibridnykh sistem”, Differentsialnye uravneniya, 44:11 (2008), 1523–1533 | MR | Zbl

[88] Kurzhanskii A.B., Filippova T.F., “Differentsialnye vklyucheniya s fazovymi ogranicheniyami. Metod vozmuschenii”, Optimalnoe upravlenie i differentsialnye uravneniya, Trudy MI RAN, 211, M., 1995, 304–315 | MR

[89] Kurilenko A.M., “Svoistva lineinykh dinamicheskikh sistem so sluchainymi parametrami”, Izvestiya AN SSSR. TK, 1984, no. 4, 183–191 | MR | Zbl

[90] Levakov A.A., “K upravlyaemosti lineinykh nestatsionarnykh sistem”, Differentsialnye uravneniya, 23:5 (1987), 798–806 | MR | Zbl

[91] Levitan B.M., Zhikov V.V., Pochti-periodicheskie funktsii i differentsialnye uravneniya, Izdatelstvo Moskovskogo universiteta, M., 1978, 205 pp. | MR | Zbl

[92] Leikhtveis K., Vypuklye mnozhestva, Nauka, M., 1985, 335 pp. | MR

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

[94] Marchuk G.I., Matematicheskoe modelirovanie v probleme okruzhayuschei sredy, Nauka, M., 1982 \, 320 pp. | MR

[95] Masterkov Yu.V., Rodina L.I., “Dostatochnye i neobkhodimye usloviya ustoichivoi upravlyaemosti nelineinoi nestatsionarnoi sistemy na ploskosti v kriticheskom sluchae”, Differentsialnye uravneniya, 39:2 (2003), 259–267 | MR

[96] Masterkov Yu.V., Rodina L.I., “Dostatochnye usloviya ustoichivoi upravlyaemosti nestatsionarnoi sistemy v kriticheskom sluchae”, Differentsialnye uravneniya, 40:1 (2004), 68–75 | MR

[97] Masterkov Yu.V., Rodina L.I., “O postroenii neuprezhdayuschego upravleniya dlya sistem so sluchainymi parametrami”, Vestnik Udmurtskogo universiteta. Matematika, 2005, no. 1, 101–114

[98] Masterkov Yu.V., Rodina L.I., “Usloviya lokalnoi upravlyaemosti sistem so sluchainymi parametrami”, Vestnik Udmurtskogo universiteta. Matematika, 2006, no. 1, 81–94

[99] Masterkov Yu.V., Rodina L.I., “Upravlyaemost lineinoi dinamicheskoi sistemy so sluchainymi parametrami”, Differentsialnye uravneniya, 43:4 (2007), 457–464 | MR | Zbl

[100] Masterkov Yu.V., Rodina L.I., “Funktsii Lyapunova upravlyaemykh sistem so sluchainymi parametrami”, Differentsialnye uravneniya, 43:6 (2007), 858–859 | MR

[101] Masterkov Yu.V., Rodina L.I., “Dostatochnye usloviya lokalnoi upravlyaemosti sistem so sluchainymi parametrami dlya proizvolnogo chisla sostoyanii sistemy”, Izvestiya vysshikh uchebnykh zavedenii. Matematika, 2008, no. 3(550), 38–49 | MR | Zbl

[102] Milshtein G.N., “Srednekvadraticheskaya ustoichivost lineinykh sistem, nakhodyaschikhsya pod vozdeistviem markovskoi tsepi”, Prikladnaya matematika i mekhanika, 36:3 (1972), 537–545 | MR

[103] Milshtein G.N., Repin Yu.M., “O vozdeistvii markovskogo protsessa na sistemu differentsialnykh uravnenii”, Differentsialnye uravneniya, 5:8 (1969), 1371–1384 | MR

[104] Minyuk S.A., “K teorii polnoi upravlyaemosti lineinykh nestatsionarnykh sistem”, Differentsialnye uravneniya, 26:3 (1990), 414–420 | MR | Zbl

[105] Nemytskii V.V., Stepanov V.V., Kachestvennaya teoriya differentsialnykh uravnenii, GITTL, M., 1949, 550 pp.

[106] Nikolaev S.F., Tonkov E.L., “Differentsiruemost vektora bystrodeistviya i pozitsionnoe upravlenie lineinoi dokriticheskoi sistemoi”, Differentsialnye uravneniya, 36:1 (2000), 76–84 | MR | Zbl

[107] Nikolaev S.F., Tonkov E.L., “O nekotorykh zadachakh, svyazannykh s suschestvovaniem i postroeniem neuprezhdayuschego upravleniya dlya nestatsionarnykh upravlyaemykh sistem”, Vestnik Udmurtskogo universiteta. Matematika. Izhevsk, 2000, no. 1, 11–32

[108] Nikolskii M.S., “Ob approksimatsii mnozhestva dostizhimosti dlya differentsialnogo vklyucheniya”, Vestnik Moskovskogo universiteta. Seriya Vychislitelnaya matematika i kibernetika, 4 (1987), 31–34 | MR

[109] Obukhovskii V.V., “O topologicheskoi stepeni dlya odnogo klassa nekompaktnykh mnogoznachnykh otobrazhenii”, Funktsionalnyi analiz (Ulyanovsk), 1984, no. 23, 82–93 | MR

[110] Ovseevich A.I., Chernousko F.L., “Nekotorye svoistva optimalnykh ellipsoidov, approksimiruyuschikh mnozhestva dostizhimosti”, Doklady AN, 388:4 (2003), 462–465 | MR | Zbl

[111] Oseledets V.I., “Markovskie tsepi, kosye proizvedeniya i ergodicheskie teoremy dlya «obschikh» dinamicheskikh sistem”, Teoriya veroyatnostei i ee primeneniya, 10:3 (1965), 551–557

[112] Osipenko G.S., “K voprosu ob approksimatsii invariantnykh mer dinamicheskikh sistem”, El. zh. Differentsialnye uravneniya i protsessy upravleniya, 2008, no. 2, 57–79 ; URL: http://www.neva.ru/journal | MR

[113] Osipenko G.S., Krupin A.V., Bezruchko A.A., Petrenko E.I., Kapitanov A.A., “Postroenie invariantnykh mer dinamicheskikh sistem”, El. zh. Differentsialnye uravneniya i protsessy upravleniya, 2007, no. 4, 27–51 ; URL: http://www.neva.ru/journal | MR

[114] Panasenko E.A., Rodina L.I., Tonkov E.L., “Pogloschaemost, nebluzhdaemost i rekurrentnost mnozhestva dostizhimosti upravlyaemoi sistemy”, Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternye nauki, 2008, no. 2, 97–104 | MR

[115] Panasenko E.A., Rodina L.I., Tonkov E.L., “Asimptoticheski ustoichivye statisticheski slabo invariantnye mnozhestva upravlyaemykh sistem”, Trudy Instituta matematiki i mekhaniki UrO RAN, 16:5 (2010), 135–142 | MR

[116] Panasenko E.A., Rodina L.I., Tonkov E.L., “Prostranstvo $\mathrm{clcv}(\mathbb R^n)$ s metrikoi Khausdorfa – Bebutova i differentsialnye vklyucheniya”, Trudy Instituta matematiki i mekhaniki UrO RAN, 17:1 (2011), 162–177

[117] Panasenko E.A., Tonkov E.L., “Funktsii Lyapunova i polozhitelno invariantnye mnozhestva differentsialnykh vklyuchenii”, Differentsialnye uravneniya, 43:6 (2007), 859–860

[118] Panasenko E.A., Tonkov E.L., “Invariantnye i ustoichivo invariantnye mnozhestva differentsialnykh vklyuchenii”, Trudy Matematicheskogo instituta im. V.A. Steklova, 262, 2008, 202–221 | MR | Zbl

[119] Panasenko E.A., Tonkov E.L., “Rasprostranenie teorem E.A. Barbashina i N.N. Krasovskogo ob ustoichivosti na upravlyaemye dinamicheskie sistemy”, Trudy Instituta matematiki i mekhaniki UrO RAN, 15:3 (2009), 185–201

[120] Perov A.I., “Neskolko zamechanii otnositelno differentsialnykh neravenstv”, Izvestiya vysshikh uchebnykh zavedenii. Matematika, 1965, no. 4(47), 104–112 | MR | Zbl

[121] Plotnikov V.A., Plotnikov A.V., Vityuk A.N., Differentsialnye uravneniya s mnogoznachnoi pravoi chastyu. Asimptoticheskie metody, AstroPrint, Odessa, 1999, 355 pp. | MR

[122] Popova S.N., Tonkov E.L., “Soglasovannye sistemy i upravlenie pokazatelyami Lyapunova”, Differentsialnye uravneniya, 33:2 (1997), 226–235 | MR | Zbl

[123] Pugachev V.S., Teoriya sluchainykh funktsii i ee primenenie k zadacham avtomaticheskogo upravleniya, Fizmatgiz, M., 1962, 884 pp. | MR

[124] Rodina L.I., “O lokalnoi upravlyaemosti sistem so sluchainymi parametrami”, Chetvertye Bogdanovskie chteniya po obyknovennym differentsialnym uravneniyam, Tezisy dokladov, Minsk, 2005, 116–117

[125] Rodina L.I., “O suschestvovanii neuprezhdayuschego upravleniya dlya sistem so sluchainymi parametrami”, Izvestiya Instituta matematiki i informatiki UdGU. Izhevsk, 2006, no. 2(36), 95–98

[126] Rodina L.I., “Usloviya suschestvovaniya neuprezhdayuschego upravleniya dlya sistem so sluchainymi parametrami”, Izvestiya Instituta matematiki i informatiki UdGU. Izhevsk, 2006, no. 3(37), 131–132

[127] Rodina L.I., “Ob asimptoticheskoi ustoichivosti s veroyatnostyu edinitsa invariantnykh mnozhestv differentsialnykh vklyuchenii so sluchainymi parametrami”, Vestnik Tambovskogo Universiteta, 12:4 (2007), 520–521 | MR

[128] Rodina L.I., “Statisticheski invariantnye s veroyatnostyu edinitsa mnozhestva upravlyaemykh sistem so sluchainymi parametrami”, Differentsialnye uravneniya, 47:6 (2011), 903–905

[129] Rodina L.I., “Statisticheski invariantnye mnozhestva upravlyaemykh sistem so sluchainymi parametrami”, Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternye nauki, 2011, no. 2, 68–87

[130] Rodina L.I., “Funktsii Lyapunova i statisticheski invariantnye mnozhestva upravlyaemykh sistem so sluchainymi parametrami”, Mezhdunarodnaya konferentsiya «Differentsialnye uravneniya i smezhnye voprosy», posvyaschennaya pamyati I.G. Petrovskogo, Tezisy dokladov, Izd-vo MGU, M., 2011, 320–321

[131] Rodina L.I., Tonkov E.L., “Kriterii polnoi upravlyaemosti lineinoi nestatsionarnoi sistemy v kriticheskom sluchae”, Izvestiya Instituta matematiki i informatiki UdGU. Izhevsk, 2002, no. 2(25), 81–86

[132] Rodina L.I., Tonkov E.L., “Usloviya polnoi upravlyaemosti nestatsionarnoi lineinoi sistemy v kriticheskom sluchae”, Kibernetika i sistemnyi analiz, 2004, no. 3, 87–100 | MR | Zbl

[133] Rodina L.I., Tonkov E.L., “Statisticheskie kharakteristiki mnozhestva dostizhimosti upravlyaemoi sistemy, nebluzhdaemost i minimalnyi tsentr prityazheniya”, Nelineinaya dinamika, 5:2 (2009), 265–288

[134] Rodina L.I., Tonkov E.L., “Statisticheski invariantnye mnozhestva upravlyaemoi sistemy”, Vestnik Tambovskogo Universiteta, 14:4 (2009), 788–790

[135] Rodina L.I., Tonkov E.L., “Statisticheski slabo invariantnye mnozhestva upravlyaemykh sistem”, Mezhdunarodnaya konferentsiya, posvyaschennaya 70-letiyu rektora MGU akademika V. A. Sadovnichego, Tezisy dokladov, Izd-vo MGU, M., 2009, 333–334

[136] Rodina L.I., Tonkov E.L., “Statisticheski invariantnye mnozhestva upravlyaemoi sistemy, parametrizovannoi dinamicheskoi sistemoi”, Mezhdunarodnaya konferentsiya po differentsialnym uravneniyam i dinamicheskim sistemam, Tezisy dokladov, Suzdal, 2010, 161–162

[137] Rodina L.I., Tonkov E.L., “Statisticheski slabo invariantnye mnozhestva upravlyaemykh sistem”, Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternye nauki, 2011, no. 1, 67–86

[138] Rodina L.I., Tonkov E.L., “O suschestvovanii statisticheski invariantnykh mnozhestv upravlyaemykh sistem so sluchainymi parametrami”, Mezhdunarodnaya konferentsiya po matematicheskoi teorii upravleniya i mekhanike, Tezisy dokladov, Suzdal, 2011, 174–177

[139] Rozanov Yu.A., Statsionarnye sluchainye protsessy, Nauka, M., 1990, 272 pp. | MR

[140] Rokhlin V.A., “Izbrannye voprosy metricheskoi teorii dinamicheskikh sistem”, Uspekhi matematicheskikh nauk, 4:2 (1949), 57–128 | MR | Zbl

[141] Rokhlin V.A., “Lektsii po entropiinoi teorii preobrazovanii s invariantnoi meroi”, Uspekhi matematicheskikh nauk, 22:5 (1967), 3–56 | Zbl

[142] Sachkov Yu.L., “Invariantnye oblasti trekhmernykh bilineinykh sistem”, Vestnik MGU. Matematika, mekhanika, 1991, no. 4, 23–26 | MR

[143] Sachkov Yu.L., “Upravlyaemost dvumernykh i trekhmernykh bilineinykh sistem v polozhitelnom ortante”, Differentsialnye uravneniya, 29:2 (1993), 361–363 | MR | Zbl

[144] Sachkov Yu.L., “Invariantnye ortanty bilineinykh sistem”, Differentsialnye uravneniya, 31:6 (1995), 1094–1095 | MR | Zbl

[145] Sinai Ya.G., “O slabom izomorfizme preobrazovanii s invariantnoi meroi”, Matematicheskii sbornik, 63:1 (1964), 23–42 | MR

[146] Sirotin A.N., “O zadache ogranichennoi nul-upravlyaemosti s veroyatnostyu 1 dlya lineinykh avtonomnykh sistem s diskretnym vremenem i sluchainoi perekhodnoi matritsei s konechnym mnozhestvom spektrov”, Avtomatika i telemekhanika, 1996, no. 11, 39–51 | MR | Zbl

[147] Smirnov E.Ya., Nekotorye zadachi matematicheskoi teorii upravleniya, Izd-vo Leningradskogo universiteta, L., 1981, 200 pp. | MR | Zbl

[148] Solodovnikov V.V., Statisticheskaya dinamika lineinykh sistem avtomaticheskogo upravleniya, Fizmatgiz, M., 1960, 470 pp. | MR

[149] Subbotin A.I., “Monotonnye otnositelno predporyadka traektorii differentsialnykh vklyuchenii”, Trudy Instituta matematiki i mekhaniki UrO RAN, 1 (1992), 138–146 | Zbl

[150] Subbotin A.I., Subbotina N.N., Tretyakov V.E., “Stokhasticheskoe i determinirovannoe upravlenie. Differentsialnye neravenstva”, Problemy upravleniya i teoriya informatsii, 14:6 (1985), 1–15 | MR

[151] Subbotin A.I., Chentsov A.G., Optimizatsiya garantii v zadachakh upravleniya, Nauka, M., 1981, 286 pp. | MR

[152] Subbotina N.N., “Metod kharakteristik dlya uravnenii Gamiltona–Yakobi i ego prilozheniya v dinamicheskoi optimizatsii”, Sovremennaya matematika i ee prilozheniya, 20 (2004), 1–133

[153] Tolstonogov A.A., Differentsialnye vklyucheniya v banakhovom prostranstve, Nauka, M., 1986, 297 pp. | MR | Zbl

[154] Tolstonogov A.A., “Svoistva mnozhestv dostizhimosti evolyutsionnykh vklyuchenii i upravlyaemykh sistem subdifferentsialnogo tipa”, Sibirskii matematicheskii zhurnal, 45:4 (2004), 920–945 | MR | Zbl

[155] Tonkov E.L., “Dinamicheskaya sistema sdvigov i voprosy ravnomernoi upravlyaemosti lineinoi sistemy”, Doklady AN SSSR, 256:2 (1981), 290–294 | MR | Zbl

[156] Tonkov E.L., “Dinamicheskaya sistema sdvigov i voprosy globalnoi upravlyaemosti lineinoi pochti periodicheskoi sistemy”, Uspekhi matematicheskikh nauk, 36:4(220) (1981), 191–241

[157] Tonkov E.L., “Veroyatnostnye kharakteristiki mnozhestva upravlyaemosti lineinogo differentsialnogo uravneniya”, Uspekhi matematicheskikh nauk, 37:4 (1982), 121

[158] Tonkov E.L., “O mnozhestve upravlyaemosti lineinogo uravneniya”, Differentsialnye uravneniya, 19:2 (1983), 269–278 | MR | Zbl

[159] Tonkov E.L., “Kanonicheskii predstavitel lineinoi upravlyaemoi sistemy”, Vestnik Udmurtskogo universiteta. Matematika. Izhevsk, 2003, no. 1, 113–128

[160] Tonkov E.L., “Globalno upravlyaemye lineinye sistemy”, Sovremennaya matematika i ee prilozheniya, 23 (2005), 145–165

[161] Ushakov V.N., Zavarin A.B., “O vydelenii yadra vyzhivaemosti dlya differentsialnogo vklyucheniya”, Prikladnaya matematika i mekhanika, 65:5 (2001), 831–842 | MR | Zbl

[162] Ushakov V.N., Latushkin Ya.A., “Defekt stabilnosti mnozhestv v igrovykh zadachakh upravleniya”, Trudy Instituta matematiki i mekhaniki UrO RAN, 12:2 (2006), 178–194

[163] Ushakov V.N., Matviichuk A.R., Lebedev P.D., “Defekt stabilnosti v igrovoi zadache o sblizhenii v moment”, Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternye nauki, 2010, no. 3, 87–103

[164] Ushakov V.N., Malev Ya.A., “K voprosu o defekte stabilnosti mnozhestv v igrovoi zadache o sblizhenii”, Trudy Instituta matematiki i mekhaniki UrO RAN, 16:1 (2010), 199–222

[165] Ushakov V.N., Zimovets A.A., “Defekt invariantnosti mnozhestv otnositelno differentsialnogo vklyucheniya”, Vestnik Udmurtskogo universita. Matematika. Mekhanika. Kompyuternye nauki, 2011, no. 2, 98–111 | MR

[166] Filippov A.F., “Differentsialnye uravneniya s razryvnoi pravoi chastyu”, Matematicheskii sbornik, 51(93):1 (1960), 99–128 | Zbl

[167] Filippov A.F., “Klassicheskie resheniya differentsialnykh uravnenii s mnogoznachnoi pravoi chastyu”, Vestnik Moskovskogo universiteta. Matematika, mekhanika, 1967, no. 3, 16–26 | Zbl

[168] Filippov A.F., “O suschestvovanii reshenii mnogoznachnykh differentsialnykh uravnenii”, Matematicheskie zametki, 10:3 (1971), 307–313 | Zbl

[169] Filippov A.F., “Usloviya ustoichivosti odnorodnykh sistem s proizvolnymi pereklyucheniyami rezhimov”, Avtomatika i telemekhanika, 1980, no. 8, 48–55 | Zbl

[170] Filippov A.F., Differentsialnye uravneniya s razryvnoi pravoi chastyu, Nauka, M., 1985, 223 pp. | MR

[171] Filippov A.F., Vvedenie v teoriyu differentsialnykh uravnenii, Editorial URSS, M., 2004, 240 pp.

[172] Fleming U., Rishel R., Optimalnoe upravlenie determinirovannymi i stokhasticheskimi sistemami, Mir, M., 1978, 316 pp. | MR

[173] Khartman F., Obyknovennye differentsialnye uravneniya, Mir, M., 1970, 720 pp. | MR

[174] Khasminskii R.Z., Ustoichivost sistem differentsialnykh uravnenii pri sluchainykh vozmuscheniyakh ikh parametrov, Nauka, M., 1969, 367 pp. | MR

[175] Chaplygin S.A., “Novyi metod priblizhennogo integrirovaniya differentsialnykh uravnenii”, Izbrannye trudy. Mekhanika zhidkosti i gaza. Matematika, Nauka, M., 1976, 307–362 | MR

[176] Chentsov A.G., “K igrovoi zadache navedeniya”, Doklady AN SSSR, 226:1 (1976), 73–76 | MR | Zbl

[177] Chentsov A.G., “Ob igrovoi zadache sblizheniya v zadannyi moment vremeni”, Matematicheskii sbornik, 99:3 (1976), 394–420 | MR | Zbl

[178] Chentsov A.G., Prilozheniya teorii mery k zadacham upravleniya, Sredn.-Ural. knizhnoe izd-vo, Sverdlovsk, 1985, 128 pp.

[179] Shiryaev A.N., Veroyatnost, Nauka, M., 1989 \, 640 pp. | MR

[180] Arnold L., Random dynamical systems publaddr Berlin–Heidelberg, Springer-Verlag, 1998, 586 pp. | MR

[181] Aubin J.P., Viability theory, Birkhäuser, Boston, 1991, 543 pp. | MR | Zbl

[182] Aubin J.P., Cellina A., Differential inclusions. Set-valued maps and viability theory, Springer-Verlag, Berlin–Heidelberg–New York–Tokyo, 1984, 342 pp. | MR

[183] Aubin J. P.,, “Da Prato G. Stochastic viability and invariance”, Annali Scuola Normale di Pisa, 1990, no. 27, 595–694 | MR

[184] Aubin J.P., Frankowska H., “Heavy viable trajectories of controlled systems”, Annales de l'Institut Henri-Poincaré. Analyse Non Linéaire, 1985, no. 2, 371–395 | MR | Zbl

[185] Basile G., Marro G., “Controlled and conditional invariant subspaces in linear system theory”, J. Optim. Theory Appl., 1969, no. 3, 296–315

[186] Bensoussan A., Lions J.L., Applications of variational inequalities in stochastic control, North-Holland Publishing Company, Amsterdam–New York–Oxford, 1982, 564 pp. | MR

[187] Booton R.C., “Nonlinear control systems with random inputs”, Trans. IRE, CT-1 (1954), 9–18

[188] Bressan A., “Upper and lower semicontinuous differential inclusions: a unified approach”, Nonlinear controllability and optimal control, Monogr. Textbooks Pure Appl. Math., 133, Dekker, New York, 1990, 21–31 | MR | Zbl

[189] Brockett R., “On the reachable set for bilinear systems”, Variable Structure Systems, Lecture Notes in Economics and Math. Systems, 111, Springer-Verlag, 1971, 54–63 | DOI | MR

[190] Chang A., “An algebraic characterization of controllability”, IEEE Trans. Autom. Control, 10:1 (1965), 112–114 | DOI

[191] Clarke F.H., “Generalized gradients and applications”, Trans. Amer. Math. Soc., 205:2 (1975), 247–262 | DOI | MR | Zbl

[192] Clarke F.H., Ledyaev Yu.S., Stern R.J., Wolenski P.R., Nonsmooth analysis and control theory, Springer, New York, 1998, 296 pp. | MR

[193] Colonius F., Jonson R., “Local and global null controllability of time varying linear control systems”, Control, Optimisation and Calculus of Variations, 2 (1997), 329–341 | DOI | MR | Zbl

[194] Conti R., Linear differential equations and control, Academic Press, London, 1976, 174 pp. | MR | Zbl

[195] Crandall M.G., “A generalisation of Peano's existence theorem and flow invariance”, Proc. Amer. Math. Soc., 36:1 (1972), 151–155 | MR | Zbl

[196] Davy J.L., “Properties of the solution set of a generalized differential equations”, Bull. Austral. Math. Soc., 6 (1972), 379–398 | DOI | MR | Zbl

[197] De Farias D.P., Geromel J.C., Do Val J.B.R., Costa O.L.V., “Output feedback control of Markov jump linear systems in continuous-time”, IEEE Trans. Autom. Control, 45:5 (2000), 944–949 | DOI | MR | Zbl

[198] Deimling K., Multivalued differential equations, Walter de Gruyter, Berlin–New York, 1992, 260 pp. | MR | Zbl

[199] Fleming W.H., Soner H.M., Controlled Markov processes and viscosity solutions, Springer-Verlag, New York, 2005, 448 pp.

[200] Frankowska H., “Local controllability of control systems with feedbacks”, Journal of Optimization Theory and Applications, 1989, no. 60, 277–296 | DOI | MR | Zbl

[201] Galperin E.A., “Some generalization of Lyapunov's approach to stability and control”, Nonlin. Dynam. and Syst. Theory, 2:1 (2002), 1–24 | MR

[202] Guseinov H.G., Subbotin A.I., Ushakov V.N., “Derivatives for multivalued mappings with applications to game theoretical problems of control”, Probl. Contr. Inform. Theory, 14:3 (1985), 155–167 | MR | Zbl

[203] Haddad G., “Monotone trajectories of differential inclusions and functional-differential inclusions with memory”, Izrael J. Math., 39:1 (1981), 83–100 | DOI | MR | Zbl

[204] Hartman P., “On invariant sets and on a theorem of Wazewski”, Proc. Amer. Math. Soc., 1972, no. 32, 511–520 | MR | Zbl

[205] Hilmy H., “Sur les centres d'attraction minimaux dans les systemes dynamiques”, Comp. Math., 3:2 (1936), 187–204 | MR

[206] Himmelberg C.G. van Vleck F.S., “Existence of solutions for generalized differential equations with unbounded right-hand side”, J. Differential Equations, 61:3 (1986), 295–320 | DOI | MR | Zbl

[207] Hu S., Papageorgiou N.S., Handbook of multivalued analysis, v. I, Theory, Kluwer, Dordrecht, 1997, 980 pp. | MR | Zbl

[208] Hu S., Papageorgiou N.S., Handbook of multivalued analysis, v. II, Applications, Kluwer, Dordrecht, 2000, 918 pp. | MR

[209] Ibrir S., Boukas E.K., “A constant-gain nonlinear estimator for linear switching systems”, Nonlin. Dynam. and Syst. Theory, 5:1 (2005), 49–59 | MR | Zbl

[210] Kalman R.E., “Contribution to the theory of optimal control”, Boletin de la Sociedad Matematika Mexicana, 5:1 (1960), 102–119 | MR | Zbl

[211] Kalman R.E., Ho Y.C., Narendra K.S., “Controllability of linear dynamical systems”, Contr. Different. Equat., 1 (1963), 189–213 | MR | Zbl

[212] Khasminsky R.Z., “Limit theorem for a solution of the differential equation with a random right part”, Prob. Theor. and its Applic., 11:3 (1966), 444–462 | MR

[213] Krylov N.M., Bogolyubov N.N., “La théorie générale de la mesure et son application à l'etude des systèmes dynamiques de la mechanique non linéaire”, Annals of Mathematics, 1:38 (1937), 65–113 | DOI | MR

[214] Kurzhanski A.B., Filippova T.F., “Dynamics of the set of viable trajectories to a differential inclusion: the evolution equation”, Probl. Contr. Inform. Theory, 17:3 (1988), 137–144 | MR | Zbl

[215] Kurzhanski A.B., Filippova T.F., “Pertubation technicues for viability and control”, Lect. Notes in Control, Inform. Sci., 180 (1992), 394–403 | DOI | MR | Zbl

[216] La Salle J.P., “Time optimal control systems”, Proc. Nat. Acad. Sci. USA, 1:45 (1959), 4–13 | MR

[217] Marchaud A., “Sur les champs de demi-cones et equations differentielles du premier ordre”, Bull. Soc. Math. France, 62 (1934), 1–38 | MR

[218] Marchaud A., “Sur les champs de demi-cones convexes”, Bull. Sci. Math., 62 (1938), 229–240

[219] Masterkov Yu.V., Rodina L.I., “The Sufficient conditions of local controllability for linear systems with random parameters”, Nonlin. Dynam. and Syst. Theory, 2007, no. 7(3), 303–314 | MR | Zbl

[220] Nagumo M., “Über die Laga der integralkurven gewöhnlicher differential Gleichungen”, Proc. Phys. Math. Japan, 24 (1942), 399–414 | MR

[221] Quincampoix M., “Differential inclusions and target problems”, SIAM J. Control and Optimizat., 30:2 (1992), 324–335 | DOI | MR | Zbl

[222] Quincampoix M., Buckdahn R., Rainer C. and Teichman J., “Another proof for the equivalence between invariance of closed sets with respect to stochastic and deterministic systems”, Bulletin des Sciences Mathematiques, 134 (2010), 207–214 | DOI | MR | Zbl

[223] Rockafellar R.T., “Generalized directional derivatives and subgradients of nonconvex functions”, Can. J. Math., 1980, no. 32, 157–180 | MR

[224] Rockafellar R.T., Wets R.J.-B., Variational analysis, Springer-Verlag, New York, 1998, 348 pp. | MR | Zbl

[225] Rodina L.I., “Conditions of total controllability of linear nonstationary systems in the critical case”, The International Conference on Applied Mathematics Dedicated to the 65-th Anniversary of B.N. Pshenichnyi (1937–2000), Abstracts, Kyiv, Ukraine, 2002, 72

[226] Rodina L.I., Tonkov E.L., The Statistical invariant sets of controllable systems, Preprints of IFAC Workshop on Control Applications of Optimisation, University of Jyvaskyla, Finland, 6–8 May 2009

[227] Roxin E., “Stability in general control systems”, Journal of Dif. Equat., 1:2 (1965), 115–150 | DOI | MR | Zbl

[228] San Martin L.A.B., “Invariant control sets on flag manifolds”, Math. Control Signals Systems, 6 (1993), 41–61 | DOI | MR | Zbl

[229] Stepanoff W., “Sur une extension du theoreme ergodique”, Comp. Math., 1936, no. 3, 68–85 | MR

[230] Tsarkov Ye., “Asymptotic methods for stability analysis of Markov impulse dynamical systems”, Nonlin. Dynam. and Syst. Theory, 2:1 (2002), 103–115 | MR | Zbl

[231] Zaremba S.K., “Sur une extension de la notion d'équation differentielle”, C. R. Acad. Sci. Paris, 199:10 (1934), 545–548

[232] Zaremba S.K., “Sur les equations au paratingent”, Bull. Sci. Math., 60:2 (1936), 139–160