Necessary optimality conditions for nonconvex differential inclusions with endpoint constraints

Journal of Differential Equations

Volumn 124, Issue 1, 1996, Pages 186-204

  Zhu, Q J ^a  

^a WESTERN MICHIGAN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0029715234     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.1996.0006     Document Type: Article

References (35)

1. 1. J. P. AUBIN AND A. CELLINA, "Differential Inclusions," Springer-Verlag, Berlin, 1984.
2. 2. F. H. CLARKE, Necessary conditions for a general control problem, in "Calculus of Variations and Control Theory" (D. Russel, Ed.), Mathematics Research Center, Pub. 36, pp. 259-278, Univ. of Wisconsin, Academic Press, New York.
3. 3. F. H. CLARKE, "Optimization and Nonsmooth Analysis," Wiley Interscience, New York, 1983.
4. 4. F. H. CLARKE, A decoupling principle in the calculus of variations, J. Math. Anal. Appl. 172 (1993), 92-105.
5. 5. N. DUNFORD AND B. J. SCHWARTZ, "Linear Operators," Vol. I, General Theory, Interscience, New York, 1958.
6. 6. A. F. FILIPPOV, Classical solutions of differential equations with multivalued right-hand side, SIAM J. Control 5 (1967), 609-621.
7. 7. H. FRANKOWSKA, The maximum principle for an optimal solution to a differential inclusion with end point constraints, SIAM J. Control Optim. 25 (1987), 145-157.
8. 8. H. FRANKOWSKA, Contingent cones to reachable sets of control systems, SIAM J. Control Optim. 27(1989), 170-198.
9. 9. A. FRYSZOWSKI AND T. RZEZUCHOWSKI, Continuous version of Filippov-Wazewski Relaxation Theorem, J. Differential Equations 94 (1991), 254-265.
10. 10. A. D. IOFFE, Single-valued representation of set-valued mappings II; application to differential inclusions, SIAM J. Control Optim. 21 (1983), 641-651.
11. 11. B. KASKOSZ, A maximum principle in relaxed controls, Nonlinear Anal., TMA 14 (1990), 357-367.
12. 12. B. KASKOSZ AND S. LOJASIEWICZ, JR., A maximum principle for generalized control systems, Nonlinear Anal., TMA 9 (1985), 109-130.
13. 13. B. KASKOSZ AND S. LOJASIEWICZ, JR., Lagrange-type extremal trajectories in differential inclusions, Systems Control Lett. 19 (1992), 241-247.
14. 14. P. D. LOEWEN AND R. T. ROCKAFELLAR, The adjoint arc in nonsmooth optimization, Trans. Amer. Math. Soc. 325 (1991), 39-72.
15. 15. P. D. LOEWEN AND R. T. ROCKAFELLAR, Optimal control of unbounded differential inclusions, SIAM J. Control Optim. 32 (1994), 442-470.
16. 16. S. LOJASIEWICZ, JR., Invariance of extremals, in "Nonlinear Controllability and Optimal Control" (H. Sussmann, Ed.), Series in Pure and Applied Math., Vol. 133, Dekker, New York, 1990.
17. 17. S. LOJASIEWICZ, JR., Parameterizations of convex sets, Progr. Approx. Theory (1991), 629-648.
18. 18. B. S. MORDUKHOVICH, Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Sov. Math. Dokl. 22 (1980), 526-530.
19. 19. B. S. MORDUKHOVICH, "Approximation Methods in Problems of Optimization and Control," Nauka, Moscow, 1988. [Russian; English transl. to appear in Wiley-Interscience]
20. 20. B. S. MORDUKHOVICH, On variational analysis in nonsmooth optimization, in "Optimization and Nonlinear Analysis" (A. Ioffe et al., Eds.), Pitman Res. Notes Math. Ser. 244 (1992), 199-213.
21. 21. B. S. MORDUKHOVICH, Discrete approximations and refined Euler-Lagrange conditions for non-convex differential inclusions, IMA Preprint Series #1115, March 1993, SIAM J. Control Optim., to appear.
22. 22. B. S. MORDUKHOVICH, Optimization and finite difference approximations of non-convex differential inclusions with free time, IMA Preprint Series #1174, September 1993, to appear in "Proceedings of the IMA workshop on Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control" (B. S. Mordukhovich and H. J. Sussmann, Eds.), IMA Volumes in Math. and Its Applications, Springer-Verlag, New York, 1995.
23. 23. B. S. MORDUKHOVICH, Necessary optimality and controllability conditions for nonsmooth control systems, in "Proc. 33rd Conf. Dec. Contr." Lake Buena Vista, FL, December 1994.
24. 24. B. S. MORDUKHOVICH, Discrete approximations of nonconvex differential inclusions, in "Proc. 33rd Conf. Dec. Contr." Lake Buena Vista, FL, December 1994.
25. 25. D. H. WAGNER, Survey of measurable selection theorems, SIAM J. Control Optim. 15 (1977), 859-903.
26. 26. J. WARGA, "Optimal Controls of Differential and Functional Equations," Academic Press, New York, 1972.
27. 27. J. WARGA, Necessary conditions without differentiability assumptions in optimal control, J. Differential Equations 18 (1975), 41-62.
28. 28. J. WARGA, Derivate containers, inverse functions, and controllability, in "Calculus of Variations and Control Theory" (D. L. Russell, Ed.), Academic Press, New York, 1976.
29. 29. J. WARGA, Fat homeomorphisms and unbounded derivate containers, J. Math. Anal. Appl. 81 (1981), 545-560.
30. 30. J. WARGA, Controllability, extremality and abnormality in nonsmooth optimal control, J. Optim. Theory Appl. 41 (1983), 239-260.
31. 31. J. WARGA, Optimization and controllability without differentiability assumptions, SIAM J. Control Optim. 21 (1983), 837-855.
32. 1 approximations, Nonlinear Anal. TMA 12 (1988), 593-597.
33. 33. J. WARGA, Global directional controllability, SIAM J. Control Optim. 27 (1989), 976-990.
34. 34. J. WARGA, An extension of the Kaskosz maximum principle, Appl. Math. Optim. 22 (1990), 61-74.
35. 35. J. WARGA AND Q. J. ZHU, The equivalence of extremals in different representations of unbounded control problems, SIAM J. Control Optim. 32 (1994), 1151-1169.