A Tikhonov-type theorem for abstract parabolic differential inclusions in Banach spaces
Anastasie Gudovich ; Mikhail Kamenski ; Paolo Nistri
Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 21 (2001), p. 207-234 / Harvested from The Polish Digital Mathematics Library
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We consider a class of singularly perturbed systems of semilinear parabolic differential inclusions in infinite dimensional spaces. For such a class we prove a Tikhonov-type theorem for a suitably defined subset of the set of all solutions for ε ≥ 0, where ε is the perturbation parameter. Specifically, assuming the existence of a Lipschitz selector of the involved multivalued maps we can define a nonempty subset ZL(ε) of the solution set of the singularly perturbed system. This subset is the set of the Hölder continuous solutions defined in [0,d], d > 0 with prescribed exponent and constant L. We show that ZL(ε) is uppersemicontinuous at ε = 0 in the C[0,d]×C[δ,d] topology for any δ ∈ (0,d].

Publié le : 2001-01-01
EUDML-ID : urn:eudml:doc:271537 
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Anastasie Gudovich; Mikhail Kamenski; Paolo Nistri. A Tikhonov-type theorem for abstract parabolic differential inclusions in Banach spaces. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 21 (2001) pp. 207-234. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1025/

[000] [1] A. Andreini, M. Kamenski and P. Nistri, A result on the singular perturbation theory for differential inclusions in Banach spaces, Topol. Methods in Nonlin. Anal. 15 (2000) 1-15. | Zbl 0971.34047

[001] [2] A. Cavallo, G. De Maria and P. Nistri, Some control problems solved via a sliding manifold approach, Diff. Eqns. and Dyn. Sys. 1 (1993), 215-230. | Zbl 0879.34048

[002] [3] J. Distel and Jr. Uhl, Vector measures, Mathematical Surveys n. 15, American Mathematical Society 1977.

[003] [4] A. Dontchev, T.Z. Donchev and I. Slavov, A Tikhonov-type theorem for singularly perturbed differential inclusions, Nonlinear Analysis TMA 26 (1996), 1547-1554. | Zbl 0852.34058

[004] [5] A. Dontchev and V.M. Veliov, Singular perturbation in Mayer's problem for linear systems, SIAM J. Control Optim. 21 (1983), 566-581. | Zbl 0519.49002

[005] [6] M. Kamenskii and P. Nistri, Periodic solutions of a singularly perturbed systems of differential inclusions in Banach spaces, in: Set-Valued Mappings with Applications in Nonlinear Analysis, Series in Mathematical Analysis and Applications 4, Gordon and Breach Science Publishers, London 2001, 213-226.

[006] [7] M. Krasnoselskii, P. Zabreiko, E. Pustyl'nik, and P. Sobolevski, Integral Operators in Spaces of Summable Functions, Noordhoff International Publishing, Leyden 1976.

[007] [8] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer Verlag New York, Inc. 1983.

[008] [9] V.M. Veliov, Differential inclusions with stable subinclusions, Nonlinear Analysis TMA 23 (1994), 1027-1038. | Zbl 0816.34011

[009] [10] V. Veliov, A generalization of the Tikhonov for singularly perturbed differential inclusions, J. Dyn. Contr. Syst. 3 (1997), 291-319. | Zbl 0943.34046