Overview of viability results; 310-321Full article in PDF format | 10.3176/phys.math.2007.4.03
Viability theorems for systems with dynamics depending on time in a measurable way, with time-dependent state constraints, are presented. We compare the results with ours using, for the first time in viability theory, generalized differential quotients. Some illustrative examples are given.
1. Nagumo, M. Uber die Lage der Intergralkurven gewōhnlicher Differentialgleichung. Proc. Phys. Math. Soc. Japan, 1942, 24, 551–559.
2. Aubin, J. P. Viability Theory. Birkhäuser, Boston, 1991.
3. Aubin, J. P. A viability approach to the inverse set-valued map theorem. J. Evol. Equations, 2006, 6, 419–432.
4. Bothe, D. Multivalued differential equations on graphs. Nonlinear Anal. Theory Methods Appl., 1992, 18, 245–252.
5. Deimling, K. Mutivalued Differential Equations. Walter de Gruyter, Berlin, New York, 1992.
6. Frankowska, H., Plaskacz, S. and Rzeÿzuchowski, T. Measurable viability theorems and the Hamilton–Jacobi–Bellman Equation. J. Differential Equations, 1995, 116, 265–305.
7. Haddad, G. Monotone trajectories of differential inclusions with memory. Isr. J. Math., 1981, 39, 83–100.
8. Haddad, G. Monotone viable trajectories for functional differential inclusions. J. Differential Equations, 1981, 42, 1–24.
9. Hu, Sh. and Papageorgiou, S. N. Handbook of Multivalued Analysis. Vol. I: Theory. Kluwer Academic Publishers, Dordrecht, 1997.
10. Hu, Sh. and Papageorgiou, S. N. Handbook of Multivalued Analysis. Vol. II: Applications. Kluwer Academic Publishers, Dordrecht, 1997.
11. Sussmann, H. J. Warga derivate containers and other generalized differentials. In Proceedings of the 41st IEEE 2002 Conference on Decision and Control. Las Vegas, Nevada, December 10–13. 2002, 1101–1106.
12. Sussmann, H. J. New theories of set-valued differentials and new version of the maximum principle of optimal control theory. In Nonlinear Control in the Year 2000 (Isidori, A., Lamnabhi-Lagarrigue, F. and Respondek, W., eds). Springer-Verlag, London, 2000, 487–526.
13. Girejko, E. On generalized differential quotients of set-valued maps. Rend. Seminario Mat. Univ. Politecnico Torino, 2005, 63, 357–362.
14. Piccoli, B. and Girejko, E. On some concepts of generalized differentials. Set-Valued Anal., 2007 (to appear).
15. Girejko, E. and Bartosiewicz, Z. Viability and generalized differential quotients. Control Cybernetics, 2006, 35, 815–830.
16. Girejko, E. and Piccoli, B. On generalized differential quotients and other generalized differentials. In On the Special Issue Control Applications of Optimisation Control and Aeronautics, Optimal Control, Control of Partial Differential Equations (Torres, D. F. M. and Trelat, E., eds), Int. J. Tomography Statistics, 2007, 5, 115–120.
17. Aubin, J. P. and Frankowska, H. Set-Valued Analysis. Birkhäuser, Boston, 1990.
18. Aubin, J. P. and Cellina, A. Differential Inclusions. Springer-Verlag, Berlin, 1984.
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