Viability and generalized differential quotients

• Autorzy: Girejko E. , Bartosiewicz Z.
• Języki publikacji: EN

Abstrakty

EN

Necessary and sufficient conditions for a set-valued map K : R → Rn to be GDQ-differentiable are given. It is shown that K is GDQ differentiate at to if and only if it has a local multiselection that is Cellina continuously approximable and Lipschitz at to. It is also shown that any minimal GDQ of K at (to,yo) is a subset of the contingent derivative of K at (to,yo), evaluated at 1. Then this fact is used to prove a viability theorem that asserts existence of a solution to the initial value problem y(t) ∈ F(t, y(t)), with y(to) =yo, where F : Gr(K) → Rn is an orientor field (i.e. multivalued vector field) defined only on the graph of K and K : T → Rn is a time-varying constraint multifunction. One of the assumptions is GDQ differentiability of K.

Słowa kluczowe

EN

viability; differential inclusion; generalized differential quotient (GDQ); contingent derivative; Cellina continuous approximability (CCA)

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2006
• Tom: Vol. 35, no 4
• Strony: 815--829
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Girejko E.

autor

Bartosiewicz Z.

• Institute of Mathematics and Physics, Bialystok Technical University, ul. Wiejska 45a, 15-351 Białystok, Poland,

Bibliografia

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• FRANKOWSKA, H., PLASKACZ, S. and RZEŻUCHOWSKI, T. (1995) Measurable viability theorems and the Hamilton-Jacobi-Bellman Journal of Differential Equations 116, 265-305.
• GIREJKO, E. (2005) On generalized differential quotients of set-valued maps. Rendiconti del Seminario Matematico dell’Universita’e del Politecnico di Torino 63 (4), 357-362.
• HU, SH. and PAPAGEORGIOU, S.N. (1997a) Handbook of Multivalued Analysis. Vol. I: Theory. Kluwer Academic Publishers, Dordrecht-Boston-London.
• HU, SH. and PAPAGEORGIOU, S.N. (1997b) Handbook of Multivalued Analysis. Vol. II: Applications, Kluwer Academic Publishers, Dordrecht-Boston-London.
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• RZEŻUCHOWSKI, T. (1980) Scorza-Dragoni type theorem for upper-semicontinuous multivalued functions. Bull. Acad. Polonaise des Science 28 (1-2), 61-65.
• SUSSMANN, H.J. (2000) New theories of set-valued differentials and new version of the maximum principle of optimal control theory. In: A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek, eds., Nonlinear Control in the Year 2000. Springer-Verlag, London, 487-526.
• SUSSMANN, H.J. (2002) Warga derivate containers and other generalized differentials, Proceedings of the 41st IEEE 2002 Conference on Decision and Control Las Vegas, Nevada, December 10-13, 1101-1106.