Equivalent subgradient versions of hamiltonian and euler-lagrange equations in variational analysis

SIAM Journal on Control and Optimization

Volumn 34, Issue 4, 1996, Pages 1300-1314

  Rockafellar, R Tyrrell ^a  

^a UNIVERSITY OF WASHINGTON   (United States)

Author keywords

Euler Lagrange equations; Hamiltonian equations; Nonsmooth analysis; Optimality; Subgradients; Variational analysis

Indexed keywords

CONSTRAINT THEORY; DIFFERENTIAL EQUATIONS; FUNCTIONS; INTEGRAL EQUATIONS; LAGRANGE MULTIPLIERS; VARIATIONAL TECHNIQUES;

EULER-LAGRANGE EQUATIONS; HAMILTONIAN EQUATIONS; NONSMOOTH ANALYSIS; VARIATIONAL ANALYSIS;

OPTIMAL CONTROL SYSTEMS;

EID: 0030191882     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012994273465     Document Type: Article

References (28)

1. R. T. ROCKAFELLAR, Convex Analysis, Princeton Univ. Press, Princeton, NJ, 1970.
2. _, Conjugate convex functions in optimal control and the calculus of variations, J. Math. Analysis Appl., 32 (1970), pp. 174-222.
3. _, Existence and duality theorems for convex problems of Bolza, Trans. Amer. Math. Soc., 159 (1971), pp. 1-40.
4. _, Generalized Hamiltonian equations for convex problems of Lagrange, Pacific J. Math., 33 (1970), pp. 411-428.
5. F. H. CLARKE, Generalized gradients and applications, Trans. Amer. Math. Soc., 205 (1975), pp. 247-262.
6. _, Optimization and Nonsmooth Analysis, Wiley, New York, 1983; reprinted as Classics in Applied Mathematics 5, Society for Industrial and Applied Mathematics, Philadelphia, 1990.
7. _, Optimization and Nonsmooth Analysis, Wiley, New York, 1983; reprinted as Classics in Applied Mathematics 5, Society for Industrial and Applied Mathematics, Philadelphia, 1990.
8. P. D. LOEWEN, Optimal Control via Nonsmooth Analysis, CRM Proc. Lecture Notes 2, American Mathematical Society, Providence, RI, 1993.
9. F. H. CLARKE, The Euler-Lagrange differential inclusion, J. Differential Equations, 19 (1975), pp. 80-90.
10. _, Extremal arcs and extended Hamiltonian systems, Trans. Amer. Math. Soc., 64 (1977), pp. 349-367.
11. _, Methods of Dynamic and Nonsmooth Optimization, CBMS-NSF Regional Conf. Ser. in Appl. Math. 57, Society for Industrial and Applied Mathematics, Philadelphia, 1989.
12. P. D. LOEWEN AND R. T. ROCKAFELLAR, The adjoint arc in nonsmooth optimization, Trans. Amer. Math. Soc., 325 (1991), pp. 39-72.
13. R. T. ROCKAFELLAR, Dualization of subgradient conditions for optimality, Nonlinear Analysis Theory Methods Appl., 20 (1993), pp. 627-646.
14. B. S. MORDUKHOVICH, Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Soviet Math. Dokl., 22 (1980), pp. 526-530.
15. _, Optimization and approximation of differential inclusions, Cybernetics, 24 (1988), pp. 781-788.
16. _, On variational analysis of differential inclusions, in Optimization and Nonlinear Analysis, A. D. Ioffe et al., eds., Longman, Harlow, 1992, pp. 199-213.
17. A. D. IOFFE, Nonsmooth subdifferentials: their calculus and applications, in Proc. 1st World Congress of Nonlinear Analysis, de Grueter, Berlin, 1995.
18. P. D. LOEWEN AND R. T. ROCKAFELLAR, Optimal control of unbounded differential inclusions, SIAM J. Control Optim., 32 (1994), pp. 442-470.
19. B. S. MORDUKHOVICH, Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions, SIAM J. Control Optim., 33 (1995), pp. 882-915.
20. A. D. IOFFE AND R. T. ROCKAFELLAR, The Euler and Weierstrass conditions for nonsmooth variational problems, Calculus of Variations and Partial Differential Equations, to appear.
21. R. J. WUSMAN, Convergence of sequences of convex sets, cones, and functions, II, Trans. Amer. Math. Soc., 123 (1966), pp. 32-45.
22. R. J.-B. WETS, Convergence of convex functions, variational inequalities, and convex optimization problems, in Variational Inequalities and Complementarity Problems, R. W. Cottle et al., eds., Wiley, New York, 1980, pp. 376-403.
23. G. SALINETTI AND R. J.-B. WETS, On the convergence of convex sets in finite dimensions, SIAM Rev. 21 (1979), pp. 18-33.
24. H. ATTOUCH, Variational Convergence of Functions and Operators, Pitman, New York, 1984.
25. J.-J. MOREAU, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), pp. 273-299.
26. R. T. ROCKAFELLAR, Extensions of subgradient calculus with applications to optimization, Nonlinear Analysis Theory Methods Appl., 7 (1985), pp. 665-698.
27. n, Nonlinear Analysis Theory Methods Appl., 3 (1979), pp. 145-154.
28. L. MCLINDEN and R. C. BERGSTROM, Preservation of convergence of convex sets and functions, Trans. Amer. Math. Soc., 268 (1981), pp. 127-142.