New necessary conditions for the generalized problem of Bolza

SIAM Journal on Control and Optimization

Volumn 34, Issue 5, 1996, Pages 1496-1511

  Loewen, P D ^a   Rockafellar, R T ^b  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

^b UNIVERSITY OF WASHINGTON   (United States)

Author keywords

Calculus of variations; Euler Lagrange condition; Hamiltonian condition; Nonsmooth analysis; Optimal control; Problem of Bolza; Transversality condition

Indexed keywords

DIFFERENTIAL EQUATIONS; FUNCTIONS; GRAPH THEORY; INTEGRATION; LAGRANGE MULTIPLIERS; OPTIMIZATION; SET THEORY;

EULER-LAGRANGE EQUATION; HAMILTONIAN EQUATION; NONSMOOTH ANALYSIS;

OPTIMAL CONTROL SYSTEMS;

EID: 0030243608     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012994275932     Document Type: Article

References (37)

1. J.-P. AUBIN, Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res., 9 (1984), pp. 87-111.
2. G. A. BLISS, Lectures on the Calculus of Variations, University of Chicago Press, Chicago, 1946.
3. F. H. CLARKE, The generalized problem of Bolza, SIAM J. Control Optim., 14 (1976), pp. 682-699.
4. _, Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983. (Republished in Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, 1991).
5. _, Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983. (Republished in Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, 1991).
6. F. H. CLARKE, Hamiltonian analysis of the generalized problem of Bolza, Trans. Amer. Math. Soc., 301 (1987), pp. 385-400.
7. _, Methods of Dynamic and Nonsmooth Optimization, CBMS-NSF Regional Conference Series 57, Society for Industrial and Applied Mathematics, Philadelphia, 1989.
8. F. H. CLARKE AND R. B. VINTER, Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Amer. Math. Soc., 289 (1985), pp. 73-98.
9. M. R. HESTENES, Calculus of Variations and Optimal Control Theory, Wiley, New York, 1966.
10. A. D. IOFFE, Approximate subdifferentials and applications I: The finite dimensional theory, Trans. Amer. Math. Soc., 281 (1984), pp. 389-416.
11. A. D. IOFFE AND R. T. ROCKAFELLAR, The Euler and Weierstrass conditions for nonsmooth variational problems, Calculus of Variations and Partial Differential Equations, 4 (1996), pp. 59-87.
12. P. D. LOEWEN, The proximal subgradient formula in Banach space, Canad. Math. Bull., 31 (1988), pp. 353-361.
13. _, Optimal Control via Nonsmooth Analysis, CRM-AMS Lecture Notes in Mathematics 2, American Mathematical Society, Providence, RI, 1993.
14. P. D. LOEWEN AND R. T. ROCKAFELLAR, The adjoint arc in nonsmooth optimization, Trans. Amer. Math. Soc., 325 (1991), pp. 39-72.
15. _, Optimal control of unbounded differential inclusions, SIAM J. Control Optim., 32 (1994), pp. 442-470.
16. B. S. MORDUKHOVICH, Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Dokl. Akad. Nauk SSSR 254 (1980), pp. 1072-1076. (In Russian; English translation in Soviet Math. Dokl., 22 (1980), pp. 526-530.)
17. B. S. MORDUKHOVICH, Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Dokl. Akad. Nauk SSSR 254 (1980), pp. 1072-1076. (In Russian; English translation in Soviet Math. Dokl., 22 (1980), pp. 526-530.)
18. _, Optimization and approximation of differential inclusions, Kibernetika 6 (1988), pp. 83-89. (In Russian; English translation in Cybernetics, 24 (1988), pp. 781-788.)
19. _, Optimization and approximation of differential inclusions, Kibernetika 6 (1988), pp. 83-89. (In Russian; English translation in Cybernetics, 24 (1988), pp. 781-788.)
20. _, Approximation Methods in Problems of Optimization and Control, Nauka, Moscow, 1988, Chapter 3.
21. _, Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions, Trans. Amer. Math. Soc., 340 (1993), pp. 1-35.
22. _, Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions, SIAM J. Control Optim., 33 (1995), pp. 882-915.
23. R. T. ROCKAFELLAR, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
24. _, Conjugate convex functions in optimal control and the calculus of variations, J. Math. Analysis Appl., 23 (1970), pp. 174-222.
25. _, Existence and duality theorems for convex problems of Bolza, Trans. Amer. Math. Soc., 159 (1971), pp. 1-40.
26. _, State constraints in convex problems of Bolza, SIAM J. Control, 10 (1972), pp. 691-715.
27. _, Existence theorems for general control problems of Bolza and Lagrange, Adv. in Math., 15 (1975), pp. 312-333.
28. _, Dual problems of Lagrange for arcs of bounded variation, in Calculus of Variations and Control Theory, D. L. Russell, ed., Academic Press, New York, 1976, pp. 155-192.
29. _, Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization, Math. Oper. Res., 6 (1981), pp. 424-436.
30. _, Extensions of subgradient calculus with applications to optimization, Nonlinear Anal., 9 (1985), pp. 665-698.
31. _, Lipschitzian properties of multifunctions, Nonlinear Anal., 9 (1985), pp. 867-885.
32. _, Hamiltonian trajectories and duality in the optimal control of linear systems with convex costs, SIAM J. Control Optim., 27 (1989), pp. 1007-1025.
33. _, Dualization of subgradient conditions for optimality, Nonlinear Anal., 20 (1993), pp. 627-646.
34. _, Equivalent subgradient versions of Hamiltonian and Euler-Lagrange equations in variational analysis, SIAM J. Control Optim., 34 (1996), pp. 1300-1314
35. G. V. SMIRNOV, Discrete approximations and optimal colutions to differential inclusions, Kibernetika, 10 (1991), pp. 76-79. (In Russian; English translation in Cybernetics, 27 (1991), pp. 101-107.)
36. G. V. SMIRNOV, Discrete approximations and optimal colutions to differential inclusions, Kibernetika, 10 (1991), pp. 76-79. (In Russian; English translation in Cybernetics, 27 (1991), pp. 101-107.)
37. R. B. VINTER AND H. ZHENG, The extended Euler-Lagrange condition for nonconvex variational problems, SIAM J. Control Optim., 35 (1997), to appear.