Geodesic
Relaxation in control systems of subdifferential type
Izvestiya. Mathematics , Tome 70 (2006) no. 1, pp. 121-152.

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In a separable Hilbert space we consider a control system with evolution operators that are subdifferentials of a proper convex lower semicontinuous function depending on time. The constraint on the control is given by a multivalued function with non-convex values that is lower semicontinuous with respect to the variable states. Along with the original system we consider the system in which the constraint on the control is the upper semicontinuous convex-valued regularization of the original constraint. We study relations between the solution sets of these systems. As an application we consider a control variational inequality. We give an example of a control system of parabolic type with an obstacle. 
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A. A. Tolstonogov. Relaxation in control systems of subdifferential type. Izvestiya. Mathematics , Tome 70 (2006) no. 1, pp. 121-152. http://geodesic.mathdoc.fr/item/IM2_2006_70_1_a5/

[1] Brezis H., Operateurs maximaux monotones et semi-groups de contractios dans les espaces de Hilbert, North-Holland, Amsterdam–London, 1973 | MR | Zbl

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

[3] Tolstonogov A. A., “Svoistva mnozhestva dopustimykh par “traektoriya-upravlenie” evolyutsionnykh upravlyaemykh sistem pervogo poryadka”, Izv. RAN. Ser. matem., 65:3 (2001), 201–224 | MR | Zbl

[4] Tolstonogov A. A., “Svoistva reshenii evolyutsionnykh upravlyaemykh sistem vtorogo poryadka. I”, Sib. matem. zhurn., 43:4 (2002), 907–923 | MR | Zbl

[5] Толстоногов А. А. “Svoistva reshenii evolyutsionnykh upravlyaemykh sistem vtorogo poryadka. II”, Sib. matem. zhurn., 43:5 (2002), 1149–1167 | MR | Zbl

[6] Himmelberg C. J., “Measurable relations”, Fundamenta Math., 87 (1975), 53–72 | MR | Zbl

[7] Burbaki N., Topologicheskie vektornye prostranstva, IL, M., 1959 | MR | Zbl

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

[9] Kenmochi N., “Solvability of nonlinear evolution equations with time-dependent constraints and applications”, Bull. Fac. Educ. Chiba University, 30 (1981), 1–87 | Zbl

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

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

[12] Tolstonogov A. A., “Svoistva mnozhestv dostizhimosti evolyutsionnykh vklyuchenii i upravlyaemykh sistem subdifferentsialnogo tipa”, Sib. matem. zhurn., 45:4 (2004), 920–945 | MR | Zbl

[13] Tolstonogov A. A., “K teoreme Skortsa–Dragoni dlya mnogoznachnykh otobrazhenii s peremennoi oblastyu opredeleniya”, Matem. zametki, 48:5 (1990), 109–120 | MR | Zbl

[14] 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

[15] Burbaki N., Obschaya topologiya. Osnovnye struktury, Nauka, M., 1968 | MR

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

[17] Hiai F., Umegaki H., “Integrals, conditional expectations and martingales of multivalued functions”, J. Multivariate Anal., 7 (1997), 149–182 | DOI | MR

[18] Fridman A., Variatsionnye printsipy i zadachi so svobodnymi granitsami, Nauka, M., 1990 | MR | Zbl

[19] Barbu V., Nonlinear semigroups and differential equations in Banach spaces, Noordhoff International Publisning, Leyden, 1976 | MR | Zbl