Find out how to access previewonly content
The Euler and Weierstrass conditions for nonsmooth variational problems
 A. D. Ioffe,
 R. T. Rockafellar
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
Necessary conditions are developed for a general problem in the calculus of variations in which the Lagrangian function, although finite, need not be Lipschitz continuous or convex in the velocity argument. For the first time in such a broadly nonsmooth, nonconvex setting, a full subgradient version of Euler's equation is derived for an arc that furnishes a local minimum in the classical weak sense, and the Weierstrass inequality is shown to accompany it when the arc gives a local minimum in the strong sense. The results are achieved through new techniques in nonsmooth analysis.
This research was supported in part by funds from the U.S.Israel Science Foundation under grant 9000455, and also by the Fund for the Promotion of Research at the Technion under grant 100954 and by the U.S. National Science Foundation under grant DMS9200303.
This article was processed by the author using the
\(LATEX\)
style filepljourlm from SpringerVerlag.
 J. Borwein, A. Ioffe: Proximal analysis on smooth spaces. SetValued Analysis (to appear)
 Borwein, J., Preiss, D. (1987) A smooth variational principle with applications. Trans. Amer. Math. Soc. 303: pp. 517527
 Clarke, F. H. (1973) Necessary conditions for nonsmooth problems of optimal control and the calculus of variations. Ph.D. thesis. University of Washington, Seattle
 Clarke, F. H. (1975) The EulerLagrange differential inclusion. J. Differ Eq. 19: pp. 8090
 Clarke, F. H. (1975) Admissible relaxation in variational and control problems. J. Math. Anal. Appl. 51: pp. 557576
 Clarke, F. H. (1976) The generalized problem of Bolza. SIAM J. Control Opt. 14: pp. 469478
 F. H. Clarke: The maximum principle under minimal hypotheses. SIAM J. Control Optim.14, 1078–1091
 Clarke, F. H. (1977) Extremal arcs and extended Hamiltonian systems. Trans. Amer. Math. Soc. 231: pp. 349367
 Clarke, F. H. (1980) The Erdmann condition and Hamiltonian inclusions in optimal control and the calculus of variations. Can. J. Math. 23: pp. 494509
 Clarke, F. H. (1983) Optimization and Nonsmooth Analysis. WileyInterscience, New York
 Clarke, F. H. (1989) Methods of Dynamic and Nonsmooth Optimization. CBMSNSF Regional conference series in applied mathematics, vol 57. SIAM Publications, Philadelphia
 Clarke, F. H. (1993) The decoupling principle in the calculus of variations. J. Math. Analysis Appl. 172: pp. 92105
 Ekeland, I., Temam, R. (1972) Convex Analysis and Variational Problems. Dunod, Paris
 Ioffe, A. (1983) On subdifferentiability spaces. Ann. New York Acad. Sci. 410: pp. 107121
 Ioffe, A. (1984) Calculus of Dini subdifferentials and contingent derivatives of setvalued maps. Nonlinear Anal. Theory Meth. Appl. 8: pp. 517539
 Ioffe, A. (1990) Proximal analysis and approximate subdifferentials. J. London Math. Soc. 41: pp. 175192
 Ioffe, A., Tikhomirov, V. (1968) Extension of variational problems. Trans. Moscow Math. Soc. 18: pp. 186246
 Ioffe, A., Tikhomirov, V. (1974) Theory of Extremal Problems. Nauka, Moscow
 Loewen, P., Rockafellar, R. T. (1991) The adjoint arc in nonsmooth optimization. Trans. Amer. Math. Soc. 325: pp. 3972
 Loewen, P., Rockafellar, R. T. (1994) Optimal control of unbounded differential inclusions. SIAM J. Control Opt. 32: pp. 442470
 Mordukhovich, B. (1988) Approximation Methods in Problems of Optimization and Control. Nauka, Moscow
 B. Mordukhovich: On variational analysis of differential inclusions, in Optimization and Nonlinear Analysis. A. Ioffe, M. Marcus, S. Reich (eds.), Pitman Research Notes in Math. vol. 244, 1992
 B. Mordukhovich: Discrete approximations and refined EulerLagrange conditions for nonconvex differential inclusions. SIAM J. Control Opt. (to appear)
 Rockafellar, R. T. (1970) Conjugate convex functions in optimal control and the calculus of variations. J. Math. Analysis Appl. 32: pp. 174222
 Rockafellar, R. T. (1970) Generalized Hamiltonian equations for convex problems of Lagrange. Pacific J. Math. 33: pp. 411428
 Rockafellar, R. T. (1971) Existence and duality theorems for convex problems of Bolza. Trans. Amer. Math. Soc. 159: pp. 140
 Rockafellar, R. T. (1993) Dualization of subgradient conditions for optimality. Nonlinear Anal. Theory Meth. Appl. 20: pp. 627646
 R. T. Rockafellar: Equivalent subgradient versions of Hamiltonian and EulerLagrange equations in variational analysis. SIAM J. Control Opt. (submitted)
 Warga, J. (1962) Relaxed variational problems. J. Math. Anal. Appl. 4: pp. 111128
 A. Ioffe: EulerLagrange and Hamiltonian formalisms in dynamic optimization (submitted)
 Title
 The Euler and Weierstrass conditions for nonsmooth variational problems
 Journal

Calculus of Variations and Partial Differential Equations
Volume 4, Issue 1 , pp 5987
 Cover Date
 19960101
 DOI
 10.1007/BF01322309
 Print ISSN
 09442669
 Online ISSN
 14320835
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 49K05
 49J52
 58C20
 Authors

 A. D. Ioffe ^{(1)}
 R. T. Rockafellar ^{(2)}
 Author Affiliations

 1. Department of Mathematics, Technion, 34608, Haifa, Israel
 2. Departments of Mathematics and Applied Mathematics, University of Washington, 98195, Seattle, USA