Article
Full entry |
PDF
(1.5 MB)
Feedback
PDF
(1.5 MB)
Feedback
Keywords:
$R_\delta $-set; homotopic; contractible; evolution triple; evolution inclusion; compact embedding; optimal control
$R_\delta $-set; homotopic; contractible; evolution triple; evolution inclusion; compact embedding; optimal control
Summary:
In the paper we study the topological structure of the solution set of a class of nonlinear evolution inclusions. First we show that it is nonempty and compact in certain function spaces and that it depends in an upper semicontinuous way on the initial condition. Then by strengthening the hypothesis on the orientor field $F(t,x)$, we are able to show that the solution set is in fact an $R_\delta $-set. Finally some applications to infinite dimensional control systems are also presented.
References:
[1] K. C. Chang: The obstacle problem and partial differential equations with discontinuous nonlinearities. Comm. Pure and Appl. Math. 33 (1980), 117–146. DOI 10.1002/cpa.3160330203 | MR 0562547 | Zbl 0405.35074
[2] F. S. DeBlasi, J. Myjak: On the solution sets for differential inclusions. Bull. Polish. Acad. Sci. 33 (1985), 17–23. MR 0798723
[3] K. Deimling, M. R. M. Rao: On solution sets of multivalued differential equations. Applicable Analysis 30 (1988), 129–135. DOI 10.1080/00036818808839797 | MR 0967566
[4] J. Dugundji: Topology. Allyn and Bacon, Inc., Boston, 1966. MR 0193606 | Zbl 0144.21501
[5] C. Himmelberg: Precompact contractions of metric uniformities and the continuity of $F(t,x)$. Rend. Sem. Matematico Univ. Padova 50 (1973), 185–188. MR 0355958
[6] C. Himmelberg, F. Van Vleck: A note on the solution sets of differential inclusions. Rocky Mountain J. Math 12 (1982), 621–625. DOI 10.1216/RMJ-1982-12-4-621 | MR 0683856
[7] D. M. Hyman: On decreasing sequences of compact absolute retracts. Fund. Math. 64 (1969), 91–97. DOI 10.4064/fm-64-1-91-97 | MR 0253303 | Zbl 0174.25804
[8] A. Lasota, J. Yorke: The generic property of existence of solutions of differential equations on Banach spaces. J. Diff. Equations 13 (1973), 1–12. DOI 10.1016/0022-0396(73)90027-2 | MR 0335994
[9] N. S. Papageorgiou: Optimal control of nonlinear evolution inclusions. J. Optim. Theory Appl. 67 (1990), 321–357. DOI 10.1007/BF00940479 | MR 1080139 | Zbl 0697.49007
[10] N. S. Papageorgiou: Convergence theorems for Banach space valued integrable multifunctions. Intern. J. Math and Math.Sci. 10 (1987), 433–442. DOI 10.1155/S0161171287000516 | MR 0896595 | Zbl 0619.28009
[11] N. S. Papageorgiou: On the solution set of differential inclusions in Banach spaces. Applicable Anal. 25 (1987), 319–329. DOI 10.1080/00036818708839695 | MR 0912190
[12] N. S. Papageorgiou: Relaxability and well-posedness for infinite dimensional optimal control problems. Problems of Control and information Theory 20 (1991), 205–218. MR 1119038 | Zbl 0741.49001
[13] L. Rybinski: On Caratheodory type selections. Fund. Math. CXXV (1985), 187–193. MR 0813756 | Zbl 0614.28005
[14] D. Wagner: Survey of measurable selection theorems. SIAM J. Control and Optim. 15 (1977), 859–903. DOI 10.1137/0315056 | MR 0486391 | Zbl 0407.28006
[15] J. Yorke: Spaces of solutions. Lecture Notes on Operations Research and Math. Economics 12 (1969), Springer, New York, 383–403. MR 0361294 | Zbl 0188.15502
[16] E. Zeidler: Nonlinear Functional Analysis and its Applications II. Springer, New York, 1990. MR 0816732 | Zbl 0684.47029
Search
Partner of
