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Nonsmooth Problems of Calculus of Variationsvia Codifferentiation∗

Published online by Cambridge University Press:  08 August 2014

Maxim Dolgopolik
Maxim Dolgopolik*
Affiliation:
Faculty of Applied Mathematics and Control Processes, Saint Petersburg State University, Petergof, 198504 Saint Petersburg, Russia. maxim.dolgopolik@gmail.com

Abstract

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In this paper multidimensional nonsmooth, nonconvex problems of the calculus of variations with codifferentiable integrand are studied. Special classes of codifferentiable functions, that play an important role in the calculus of variations, are introduced and studied. The codifferentiability of the main functional of the calculus of variations is derived. Necessary conditions for the extremum of a codifferentiable function on a closed convex set and its applications to the nonsmooth problems of the calculus of variations are described. Necessary optimality conditions in the main problem of the calculus of variations and in the problem of Bolza in the nonsmooth case are derived. Examples comparing presented results with other approaches to nonsmooth problems of the calculus of variations are given.

Keywords

Nonsmooth analysiscalculus of variationscodifferentiable functionproblem of Bolza

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 4 , October 2014, pp. 1153 - 1180
Copyright
© EDP Sciences, SMAI, 2014

References

R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
J.-P. Aubin and H. Frankowska, Set-valued analysis. Birkhauser, Boston (1990).
Bagirov, A.M., Nazari Ganjehlou, A., Ugon, J. and Tor, A.H., Truncated codifferential method for nonsmooth convex optimization. Pacific. J. Optim. 6 (2010) 483–496. Google Scholar
Bagirov, A.M. and Ugon, J., Codifferential method for minimizing DC functions. J. Glob. Optim. 50 (2011) 3–22. Google Scholar
Clarke, F.H., The generalized problem of Bolza. SIAM J. Control Optim. 14 (1976) 469–478. Google Scholar
Clarke, F.H., The Erdmann condition and Hamiltonian inclusions in optimzal control and the calculus of variations. Can. J. Math. 23 (1980) 494–509. Google Scholar
F.H. Clarke, Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983).
B. Dacorogna, Direct Methods in the Calculus of Variations. Springer Science+Business Media, LCC, New York (2008).
Demyanov, V.F., On codifferentiable functions. Vestn. Leningr. Univ., Math. 21 (1988) 27–33. Google Scholar
V.F. Demyanov, Continuous generalized gradients for nonsmooth functions, in Lect. Notes Econ. Math. Systems, edited by A. Kurzhanski, K. Neumann and D. Pallaschke, vol. 304. Springer Verlag, Berlin (1988) 24–27.
V.F. Demyanov and A.M. Rubinov, Constructive Nonsmooth Analysis. Peter Lang, Frankfurt am Main (1995).
Demyanov, V.F., Bagirov, A.M. and Rubinov, A.M., A method of truncated codifferential with application to some problems of cluster analysis. J. Glob. Optim. 23 (2002) 63–80. Google Scholar
Dolgopolik, M.V., Codifferential Calculus in Normed Spaces. J. Math. Sci. 173 (2011) 441–462. Google Scholar
Ioffe, A.D., Euler–Lagrange and Hamiltonian formalism in dynamic optimization. Trans. Amer. Math. Soc. 349 (1997) 2871–2900. Google Scholar
Ioffe, A.D. and Rockafellar, R.T., The Euler and Weierstrass conditions for nonsmooth variational problems. Calc. Var. Partial Differ. Equ. 4 (1996) 59–87. Google Scholar
A.D. Ioffe and V.M. Tihomirov, Theory of Extremal Problems. North-Holland, Amsterdam (1979).
Loewen, P.D. and Rockafellar, R.T., New Necessary Conditions for the Generalized Problem of Bolza. SIAM J. Control Optim. 34 (1996) 1496–1511. Google Scholar
B. Mordukhovich, On variational analysis of differential inclusions, in Optimization and Nonlinear Analysis, edited by A. Ioffe, M. Marcus and S. Reich, vol. 244. Pitman Res. Notes Math. Ser. Longman, Harlow, Essex (1992) 199–214.
Mordukhovich, B., Discrete approximations and refined Euler–Lagrange conditions for nonconvex differential inclusions. SIAM J. Control Optim. 33 (1995) 882–915. Google Scholar
Pallaschke, D. and Urbański, R., Reduction of quasidifferentials and minimal representations. Math. Program. 66 (1994) 161–180. Google Scholar
Rockafellar, R.T., Conjugate convex functions in optimal control and the calculus of variations. J. Math. Anal. Appl. 32 (1970) 174–222. Google Scholar
Rockafellar, R.T., Generalized Hamiltonian equations for convex problems of Lagrange. Pacific. J. Math. 33 (1970) 411–428. Google Scholar
Rockafellar, R.T., Existence and duality theorems for convex problems of Bolza. Trans. Amer. Math. Soc. 159 (1971) 1–40. Google Scholar
Scholtes, S., Minimal pairs of convex bodies in two dimensions. Mathematika 39 (1992) 267–273. Google Scholar
Vinter, R. and Zheng, H., The Extended Euler–Lagrange Condition for Nonconvex Variation Problems. SIAM J. Control Optim. 35 (1997) 56–77. Google Scholar
R. Vinter, Optimal Control. Birkhauser, Boston (2000).
K. Yosida, Functional Analysis. Springer-Verlag, New York (1980).