Geodesic
The bang-bang principle for controlled systems of subdifferential type
Trudy Instituta matematiki i mehaniki, Dynamical systems and control problems, Tome 11 (2005) no. 1, pp. 189-200 Cet article a éte moissonné depuis la source Math-Net.Ru

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In a separable Hilbert space, a nonlinear evolutionary controlled system with evolutionary operators which are subdifferentials of a proper, convex, and lower semicontinuous function depending on time is considered. The control is subjected to a constraint which is a convex closed bounded set from a separable reflexive Banach space. It is shown that any trajectory of the controlled system corresponding to a measurable control can be approximated, to any degree of accuracy and uniformly in time, by trajectories corresponding to step controls whose values are extreme points of the constraint on control. The obtained results are illustrated by an example of a controlled system with a discontinuous nonlinearity. 
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A. A. Tolstonogov. The bang-bang principle for controlled systems of subdifferential type. Trudy Instituta matematiki i mehaniki, Dynamical systems and control problems, Tome 11 (2005) no. 1, pp. 189-200. http://geodesic.mathdoc.fr/item/TIMM_2005_11_1_a16/

[1] Brézis H., Opérateursmaximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematical Studies, 5, North-Holland, Amsterdam, 1973, Notas de Matemática (50) | MR | Zbl

[2] Burbaki N., Topologicheskie vektornye prostranstva, IL, M., 1959

[3] Tolstonogov A. A., Differentsialnye vklyucheniya v banakhovom prostranstve, Nauka, Novosibirsk, 1986 | MR | Zbl

[4] Edvards R. E., Funktsionalnyi analiz: teoriya i prilozheniya, Mir, M., 1969

[5] Tolstonogov A. A., “Relaksatsiya v nevypuklykh zadachakh optimalnogo upravleniya, opisyvaemykh evolyutsionnymi uravneniyami pervogo poryadka”, Mat. sb., 190:11 (1999), 135–160 | MR | Zbl

[6] Tolstonogov A. A., Tolstonogov D. A., “$L_p$-continuous extreme selectors of multifunctions with decomposable values: relaxation theorems”, Set-Valued Anal., 4 (1996), 237–269 | DOI | MR | Zbl

[7] Tolstonogov A. A., Tolstonogov D. A., “$L_p$-continuous extreme selectors of multifunctions with decomposable values: existence theorems”, Set-Valued Anal., 4 (1996), 173–203 | DOI | MR | Zbl

[8] Kuratovskii K., Topologiya, T. 1, Mir, M., 1966 | MR

[9] Kamke E., Integral Lebega–Stiltesa, Gos. izd-vo fiz.-mat. lit., M., 1959

[10] Kenmochi N., “On the quasi-linear heat equation with time-dependent obstacles”, Nonlinear Anal., Theory, Meth., Appl., 5:1 (1981), 71–80 | DOI | MR | Zbl

[11] Tolstonogov A. A., “Approksimatsiya mnozhestv dostizhimosti evolyutsionnogo vklyucheniya subdifferentsialnogo tipa”, Sib. mat. zhurn., 44:4 (2003), 883–904 | MR | Zbl

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

[13] Filippov A. F., “Klassicheskie resheniya differentsialnykh uravnenii s mnogoznachnoi pravoi chastyu”, Vestn. MGU. Ser. Matematika i mekhanika, 1967, no. 3, 16–26 | Zbl