Measure-valued almost periodic functions

L. I. Danilov

Matematičeskie zametki, Tome 61 (1997) no. 1, pp. 57-68.

Résumé

We consider Stepanov almost periodic functions $\mu\in S(\mathbb R,\mathscr M)$ ranging in the metric space $\mathscr M$ of Borel probability measures on a complete separable metric space $\mathscr U$ is equipped with the Prokhorov metric). The main result is as follows: a function $t\to\mu[\cdot;t]\in\mathscr M$, $t\in\mathbb R$, belongs to $S(\mathbb R,\mathscr M)$ if and only if for each bounded continuous function $\mathscr F\in C_b(\mathscr U,\mathbb R)$, the function $\int_{\mathscr U}\mathscr F(x)\mu[dx;\cdot]$ is Stepanov almost periodic (of order 1) and $$ \operatorname{Mod}\mu=\sum_{\mathscr F\in C_b(\mathscr U,\mathbb R)}\operatorname{Mod}\int_{\mathscr U}\mathscr F(x)\mu[dx;\cdot]. $$

Bibliographie

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

[2] Krasovskii N. N., Upravlenie dinamicheskoi sistemoi, Nauka, M., 1985 | MR

[3] Chentsov A. G., Prilozhenie teorii mery k zadacham upravleniya, Preprint, Sverdlovsk, 1985

[4] Ivanov A. G., Meroznachnye pochti periodicheskie funktsii, Preprint, Sverdlovsk, 1990

[5] Ivanov A. G., “O pochti periodicheskoi lyapunovskoi zadache”, PMM, 55:5 (1991), 718–724 | MR | Zbl

[6] Ivanov A. G., “Optimalnoe upravlenie pochti periodicheskimi dvizheniyami”, PMM, 56:5 (1992), 133–142

[7] Danilov L. I., “O meroznachnykh pochti periodicheskikh funktsiyakh”, Vestn. Udmurtskogo un-ta, 1993, no. 1, 51–58 | Zbl

[8] Danilov L. I., Mnogoznachnye pochti periodicheskie otobrazheniya i ikh secheniya, Dep. VINITI No 2465-V93

[9] Danilov L. I., Ivanov A. G., “K teoreme o potochechnom maksimume v pochti periodicheskom sluchae”, Izv. vuzov. Matem., 1994, no. 6, 50–59 | MR | Zbl

[10] Danilov L. I., “O mnogoznachnykh pochti periodicheskikh otobrazheniyakh, zavisyaschikh ot parametra”, Vestn. Udmurtskogo un-ta, 1994, no. 2, 29–44 | Zbl

[11] Levitan B. M., Pochti-periodicheskie funktsii, GITTL, M., 1953

[12] Levitan B. M., Zhikov V. V., Pochti periodicheskie funktsii i differentsialnye uravneniya, Izd-vo MGU, M., 1978 | MR | Zbl

[13] Lyusternik L. A., Sobolev V. I., Kratkii kurs funktsionalnogo analiza, Vysshaya shkola, M., 1982 | MR | Zbl

[14] Vakhaniya N. N., Tarieladze V. I., Chobanyan S. A., Veroyatnostnye raspredeleniya v banakhovykh prostranstvakh, Nauka, M., 1985 | MR | Zbl

[15] Danilov L. I., O superpozitsii pochti periodicheskikh mnogoznachnykh otobrazhenii i funktsii, Dep. VINITI No 262-V95

[16] Danilov L. I., “Pochti periodicheskie secheniya mnogoznachnykh otobrazhenii”, Izv. otdela matem. i inform. UdGU, 1993, no. 1, 16–78 | Zbl

[17] Kelli Dzh. L., Obschaya topologiya, Nauka, M., 1981 | MR