ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

17. Control Theory and Optimization

1991 MS Classification: 26, 49, 54 Lopes-Pinto, António, University of Lisbon, Portugal, Mendes, Diana, Institute of Management and Labour Studies, Portugal Representation of Multivalued Maps The problem of representation of set-valued mappings is solved in the framework of the lower semicontinuity. “Let X, Y and Z be topological spaces, f : X × Y → Z be a separately continuous single-valued function and Γ : X → 2^{Z}\{∅} be a set-valued mapping. Assume f(x, Y) ∩ Γ(x) ≠ ∅, x ∈ X. There exists a lower semicontinuous set-valued function H : X → 2^{Y} \{∅} such that

f(x, H(s)) ⊆ Γ(x), x ∈ X?′′

This question deals directly with the lower semicontinuity criteria for the regulation map in Control Theory and Viability Theory. References:

[1] Aubin, Jean-Pierre – Viability Theory. Birkhäuser, Boston, 1991.

[2] Aubin, Jean-Pierre & Cellina, Arrigo – Differential Inclusions. Springer-Verlag, Berlin, 1984.

[3] Aubin, Jean-Pierre & Frankowska, Hélène – Set-Valued Analysis. Birkhäuser, Boston, 1984.

[4] Ekeland & Valadier – Representation on Set-Valued mappings. J. M. An. Appl., 35, 621–629 (1971)

[5] Lopes-Pinto, António – On the lower semicontinuity of multivalued maps defined implicitly, submitted.

[6] Ricceri, Biaggio – Sur la Semi-continuité Inférieure de Certaines Multifonctions. C. R. Acad. Sc. Paris, Série I, 294, 265–267 (1982).