| Title:
|
Evolution inclusions of the subdifferential type depending on a parameter (English) |
| Author:
|
Kandilakis, Dimitrios |
| Author:
|
Papageorgiou, Nikolaos S. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
33 |
| Issue:
|
3 |
| Year:
|
1992 |
| Pages:
|
437-449 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we study evolution inclusions generated by time dependent convex subdifferentials, with the orientor field $F$ depending on a parameter. Under reasonable hypotheses on the data, we show that the solution set $S(\lambda )$ is both Vietoris and Hausdorff metric continuous in $\lambda \in \Lambda $. Using these results, we study the variational stability of a class of nonlinear parabolic optimal control problems. (English) |
| Keyword:
|
subdifferential |
| Keyword:
|
compact type |
| Keyword:
|
Vietoris topology |
| Keyword:
|
Hausdorff metric |
| Keyword:
|
parabolic optimal control problem |
| MSC:
|
34A60 |
| MSC:
|
34G20 |
| MSC:
|
49A20 |
| MSC:
|
49J15 |
| MSC:
|
49J24 |
| idZBL:
|
Zbl 0783.34048 |
| idMR:
|
MR1209286 |
| . |
| Date available:
|
2009-01-08T17:57:05Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118512 |
| . |
| Reference:
|
[1] Aubin J.-P., Cellina A.: Differential Inclusions.Springer, Berlin, 1983. Zbl 0538.34007, MR 0755330 |
| Reference:
|
[2] Barbu V.: Nonlinear Semigroups and Differential Equations in Banach Spaces.Noordhoff International Publishing, Leiden, The Netherlands, 1976. Zbl 0328.47035, MR 0390843 |
| Reference:
|
[3] Brezis H.: Operateurs Maximaux Monotones.North Holland, Amsterdam, 1973. Zbl 0252.47055 |
| Reference:
|
[4] DeBlasi F., Myjak J.: On continuous approximations for multifunctions.Pacific J. Math. 123 (1986), 9-31. MR 0834135 |
| Reference:
|
[5] Hiai F., Umgaki H.: Integrals, conditional expectations and martingales of multivalued functions.J. Multiv. Anal. 7 (1977), 149-182. MR 0507504 |
| Reference:
|
[6] Klein E., Thompson A.: Theory of Correspondences.Wiley, New York, 1984. Zbl 0556.28012, MR 0752692 |
| Reference:
|
[7] Lim T.-C.: On fixed point stability for set-valued contractive mappings with applications to generalized differential equations.J. Math. Anal. Appl. 110 (1985), 436-441. Zbl 0593.47056, MR 0805266 |
| Reference:
|
[8] Nadler S.B.: Multivalued contraction mappings.Pacific J. Math. 30 (1969), 475-488. MR 0254828 |
| Reference:
|
[9] Papageorgiou N.S.: A stability result for differential m inclusions in Banach spaces.J. Math. Anal. Appl. 118 (1986), 232-246. MR 0849457 |
| Reference:
|
[10] Papageorgiou N.S.: Convergence theorems for Banach space valued integrable multifunctions.International J. Math. and Math. Sci. 10 (1987), 433-442. Zbl 0619.28009, MR 0896595 |
| Reference:
|
[11] Papageorgiou N.S.: On measurable multifunctions with applications to random multivalued equations.Math. Japonica 34 (1989), 287-296. |
| Reference:
|
[12] Papageorgiou N.S.: Infinite dimensional control systems with state and control constraints.Proceedings of the Indian Academy of Sciences 100 (1990), 65-77. Zbl 0703.49018, MR 1051092 |
| Reference:
|
[13] Papageorgiou N.S.: On evolution inclusions associated with time dependent convex subdifferentials.Comment. Math. Univ. Carolinae 31 (1990), 517-527. Zbl 0711.34076, MR 1078486 |
| Reference:
|
[14] Przyluski K.M.: Remarks on continuous dependence of an optimal control on parameters.in: Game Theory and Mathematical Economics, eds. O. Moeschlin and D. Pallashke, North Holland, Amsterdam, 1981, pp. 331-327. Zbl 0475.49025 |
| Reference:
|
[15] Rybinski L.: A fixed point approach in the study of the solution sets of Lipschitzian functional-differential inclusions.J. Math. Anal. Appl. 160 (1991), 24-46. Zbl 0735.34016, MR 1124074 |
| Reference:
|
[16] Stassinopoulos G., Vinter R.: Continuous dependence of a differential inclusion on the right-hand side with applications to stability of optimal control problems.SIAM J. Control and Optim. 17 (1979), 432-449. MR 0528904 |
| Reference:
|
[17] Tolstonogov A.: On the dependence on parameter of a solution of a differential inclusion with nonconvex second member.Diff. Equations 18 (1982), 1105-1113. MR 0672160 |
| Reference:
|
[18] Tsukada M.: Convergence of best approximations in smooth Banach spaces.J. Approx. Theory 40 (1984), 301-309. MR 0740641 |
| Reference:
|
[19] Vasilev A.: Continuous dependence of the solutions of differential inclusions on the parameter.Ukrainian Math. J. 35 (1983), 520-524. MR 0723122 |
| Reference:
|
[20] Wagner D.: Survey of measurable selection theorems.SIAM J. Control Optim. 15 (1977), 859-903. Zbl 0407.28006, MR 0486391 |
| Reference:
|
[21] Watanabe J.: On certain nonlinear evolution equations.J. Math. Soc. Japan 25 (1973), 446-463. Zbl 0253.35053, MR 0326522 |
| Reference:
|
[22] Yamada Y.: On evolution inclusions generated by subdifferential operators.J. Fac. Sci. Univ. Tokyo 23 (1976), 491-515. MR 0425701 |
| Reference:
|
[23] Yotsutani S.: Evolution equations associated with the subdifferentials.J. Math. Soc. Japan 31 (1978), 623-646. Zbl 0405.35043, MR 0544681 |
| . |
