Approximate invariance and differential inclusions in Hilbert spaces

Journal of Dynamical and Control Systems

Volumn 3, Issue 4, 1997, Pages 493-518

  Clarke, F H ^a   Ledyaev, Yu S ^b   Radulescu, M L ^c  

^a UNIVERSITÉ DE MONTRÉAL   (Canada)

^b STEKLOV MATHEMATICAL INSTITUTE   (Russian Federation)

^c UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

Approximate weak and strong invariance; Differential inclusion; Lower and upper Hamiltonians; Proximal aiming; Proximal normal cone; Viability; trajectory

Indexed keywords

APPROXIMATION THEORY; CONTROL SYSTEM ANALYSIS; SET THEORY;

DIFFERENTIAL INCLUSIONS; HILBERT SPACES;

INVARIANCE;

EID: 0031246018     PISSN: 10792724     EISSN: None     Source Type: Journal    
DOI: 10.1007/bf02463280     Document Type: Article

References (34)

1. J.-P. Aubin, Viability theory. Birkhäuser, Boston, 1991.
2. J.-P. Aubin and A. Cellina, Differential Inclusions. Springer-Verlag, New York, 1984.
3. J.-M. Bony, Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier Grenoble 19 (1969), 277-304.
4. J. M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions. Trans. Am. Math. Soc. 303 (1987), 517-527.
5. J. M. Borwein and H. M. Strojwas, Proximal analysis and boundaries of closed sets in Banach space. Part I: Theory. Can. J. Math. 38 (1986), 431-452.
6. O. Cârja and I. I. Vrabie, Some new viability results for semilinear differential inclusions. Preprint, 1996.
7. F. H. Clarke, Generalized gradients and applications. Trans. Am. Math. Soc. 205 (1975), 247-262.
8. _, Methods of dynamic and nonsmooth optimization. In: CBMS-NSF Regional Conference series in Applied Mathematics, Vol. 57. SIAM, Philadelphia, 1989.
9. _, Optimization and nonsmooth analysis. In: Classics in Applied Mathematics, Vol. 5. SIAM, Philadelphia, 1990. (Originally published by Wiley Interscience, New York, 1983).
10. F. H. Clarke and J.-P. Aubin, Monotone invariant solutions to differential inclusions. J. London Math. Soc. 16 (1977), 357-366.
11. F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Introduction to nonsmooth analysis, (textbook in preparation).
12. _, Qualitative properties of trajectories of control systems: A survey. J. Dynam. and Control Syst. 1 (1995), 1-48.
13. F. H. Clarke, Yu. S. Ledyaev, and P. R. Wolenski. Proximal analysis and minimization principles. J. Math. Anal. Appl. 196 (1995), 722-735.
14. M. G. Crandall, A generalization of Peano's existence theorem and flow invariance. Proc. Am. Math. Soc. 36 (1972), 151-155.
15. M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 277 (1983), 1-42.
16. K. Deimling, Ordinary differential equations in Banach spaces. Lect. Notes Math. 596 (1977).
17. _, Multivalued differential equations, de Gruyter, Berlin, 1992.
18. J. Diestel and J. J. Uhl, Vector measures. Am. Math. Soc., Providence, 1977.
19. H. Frankowska, Lower semicontinuous solutions of the Hamilton-Jacobi equation. SIAM J. Control Optimiz. 31 (1993), 257-272.
20. Kh. Guseinov, A. I. Subbotin, and V. N. Ushakov, Derivatives for multivalued mappings with applications to game-theoretical problems of control. Probl. Control. Inf. Theory 14 (1985), 155-167.
21. Kh. Guseinov and V. N. Ushakov, Strongly and weakly invariant sets with respect to differential inclusions. (Russian) Differ. Uravneniya 26 (1990), 1888-1894.
22. N. N. Krasovskii and A. I. Subbotin, Positional differential games. (Russian) Nauka, Moscow, 1974. (Revised English translation: Gametheoretical control problems, Springer-Verlag, New York, 1988).
23. N. N. Krasovskii and A. I. Subbotin, Positional differential games. (Russian) Nauka, Moscow, 1974. (Revised English translation: Gametheoretical control problems, Springer-Verlag, New York, 1988).
24. M. Krastanov, Ordinary forward invariant sets, homogeneity and smalltime local controllability. In: Banach Center Publications, Vol. 32. Polish Academy of Sciences, 1995.
25. P. D. Loewen, Optimal control via nonsmooth analysis. Vol. 2. CRM Proc. and Lect. Notes, Providence, Am. Math. Soc., 1993.
26. N. H. Pavel, Differential equations, flow invariance and applications. Vol. 113. Research Notes in Math., Pitman, 1984.
27. R. H. Jr. Martin, Nonlinear operators and differential equations in Banach spaces. Wiley, New York, 1976.
28. M. L. Radulescu and F. H. Clarke, Geometric approximation of proximal normals, (submitted).
29. R. Redheffer and W. Walter, A differential inequality for the distance function in normed linear spaces. Math. Ann. 211 (1974), 299-314.
30. Shuzhong Shi, Viability theorems of a class of differential-operator inclusions. J. Differ. Eqs. 79 (1989), 232-257.
31. A. I. Subbotin, A generalization of the basic equation of the theory of differential games. Sov Math. Dokl. 22 (1980), 358-362.
32. _, Generalized solutions of first-order PDEs: The dynamical optimization perspective. Birkhäuser, Boston, 1995.
33. V. M. Veliov, Sufficient conditions for viability under imperfect measurement. Set-Valued Anal. 1 (1993), 305-317.
34. J. A. Yorke, A continuous differential equation in Hilbert space without existence. Funkc. Ekvac. 13 (1970), 19-21.