Differential inclusions with state constraints. (English) Zbl 0704.49009

Papageorgiou, Nikolaos S.

Proc. Edinb. Math. Soc., II. Ser. 32, No. 1, 81-98 (1989).

This paper is devoted to the study of the semilinear nonautonomous evolution variational inequalities \[ -\dot x(t)\in N_{K(t)}(x(t))+F(t,x(t))\quad a.e.\quad [0,T],\quad x(0)=x_ 0, \] where \(N_{K(t)}(\cdot)\) is the normal cone to the convex subset K(t) and F(t,x) is a multivalued perturbation satisfying some measurability, semicontinuity and growth assumptions.
The results proved by the author cover the finite dimensional and the Hilbert space case, as well as the case of random evolution inclusion. The well posedness with respect to the initial data \(x_ 0\) and tr F is also examined.

Reviewer: P.Neittaanmäki

MSC:

49J24	Optimal control problems with differential inclusions (existence) (MSC2000)
49J40	Variational inequalities
49J55	Existence of optimal solutions to problems involving randomness
34A60	Ordinary differential inclusions

Keywords:

differential variational inequality; evolution variational inequalities; random evolution inclusion

References:

[1]	DeBlasi, Bull. Polish Acad. Sci. Math. 33 pp 17– (1985)
[2]	Dunford, Linear Operators I (1958) · Zbl 0056.34601
[3]	Daures, Sem. Anal. Convexe 8 (1974)
[4]	Delahaye, Math. Programming Study 10 pp 8– (1979) · doi:10.1007/BFb0120838
[5]	Clarke, Optimization and Nonsmooth Analysis (1984)
[6]	Cesari, Appl Math. 17 (1983)
[7]	DOI: 10.1007/BF02760619 · Zbl 0542.47036 · doi:10.1007/BF02760619
[8]	DOI: 10.1137/0315056 · Zbl 0407.28006 · doi:10.1137/0315056
[9]	Castaing, C.R. Acad Sci. Paris 282 pp 515– (1976)
[10]	Tsokos, Random Integral Equations with Applications to Life Sciences and Engineering (1974) · Zbl 0287.60065
[11]	Castaing, Convex Analysis and Measurable Multifunctions 580 (1977) · doi:10.1007/BFb0087685
[12]	Brezis, Operateurs Maximaux Monotones 5 (1973)
[13]	DOI: 10.1137/1021002 · Zbl 0421.52003 · doi:10.1137/1021002
[14]	DOI: 10.1016/0022-0396(80)90090-X · Zbl 0418.34017 · doi:10.1016/0022-0396(80)90090-X
[15]	Rockafellar, Convex Analysis (1970) · Zbl 0932.90001 · doi:10.1515/9781400873173
[16]	Aubin, Differential Inclusions (1984) · doi:10.1007/978-3-642-69512-4
[17]	DOI: 10.2307/2046246 · Zbl 0606.60058 · doi:10.2307/2046246
[18]	DOI: 10.1007/BFb0098804 · doi:10.1007/BFb0098804
[19]	Papageorgiou, Kobe J. Math. 5 pp 29– (1988)
[20]	DOI: 10.1155/S0161171287000516 · Zbl 0619.28009 · doi:10.1155/S0161171287000516
[21]	DOI: 10.1080/00036818608839594 · Zbl 0598.34010 · doi:10.1080/00036818608839594
[22]	DOI: 10.1155/S0161171286000595 · Zbl 0611.34053 · doi:10.1155/S0161171286000595
[23]	DOI: 10.1016/0022-247X(86)90305-7 · Zbl 0594.34016 · doi:10.1016/0022-247X(86)90305-7
[24]	DOI: 10.1016/0022-0396(86)90021-5 · Zbl 0615.34006 · doi:10.1016/0022-0396(86)90021-5
[25]	Nagumo, Proc. Phys-Math. Soc. Japan 24 pp 551– (1942)
[26]	DOI: 10.1016/0022-0396(77)90085-7 · Zbl 0356.34067 · doi:10.1016/0022-0396(77)90085-7
[27]	Lojasiewicz, Bull. Polish Acad. Sci. Math. 28 pp 483– (1980)
[28]	Ladde, Random Differential Inequalities (1980)
[29]	DOI: 10.1016/0022-247X(83)90032-X · Zbl 0558.34011 · doi:10.1016/0022-247X(83)90032-X
[30]	Kravvaritis, J. Differential Equations 75 (1988)
[31]	Kaczynski, Ann. Polon. Math. 29 pp 61– (1974)
[32]	Himmelberg, Fund. Math 87 pp 51– (1975)
[33]	DOI: 10.1016/0022-247X(73)90192-3 · Zbl 0262.49019 · doi:10.1016/0022-247X(73)90192-3
[34]	DOI: 10.1016/0047-259X(77)90037-9 · Zbl 0368.60006 · doi:10.1016/0047-259X(77)90037-9
[35]	Gamal, Sem. Anal. Convexe 14 (1981)
[36]	Fryszkowski, Studia Math. 76 pp 163– (1983)
[37]	DOI: 10.1016/0022-247X(85)90126-X · Zbl 0574.54012 · doi:10.1016/0022-247X(85)90126-X