Nonsmooth Problems of Calculus of Variations via Codifferentiation

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1. [1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). Zbl1098.46001MR450957
2. [2] J.-P. Aubin and H. Frankowska, Set-valued analysis. Birkhauser, Boston (1990). Zbl1168.49014MR1048347
3. [3] A.M. Bagirov, A. Nazari Ganjehlou, J. Ugon and A.H. Tor, Truncated codifferential method for nonsmooth convex optimization. Pacific. J. Optim.6 (2010) 483–496. Zbl1203.90122MR2743039
4. [4] A.M. Bagirov and J. Ugon, Codifferential method for minimizing DC functions. J. Glob. Optim.50 (2011) 3–22. Zbl1242.90172MR2787550
5. [5] F.H. Clarke, The generalized problem of Bolza. SIAM J. Control Optim.14 (1976) 469–478. Zbl0333.49023MR412926
6. [6] F.H. Clarke, The Erdmann condition and Hamiltonian inclusions in optimzal control and the calculus of variations. Can. J. Math.23 (1980) 494–509. Zbl0461.49007MR571941
7. [7] F.H. Clarke, Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983). Zbl0582.49001MR709590
8. [8] B. Dacorogna, Direct Methods in the Calculus of Variations. Springer Science+Business Media, LCC, New York (2008). Zbl1140.49001MR2361288
9. [9] V.F. Demyanov, On codifferentiable functions. Vestn. Leningr. Univ., Math. 21 (1988) 27–33. Zbl0671.49014MR965096
10. [10] 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. MR1120086
11. [11] V.F. Demyanov and A.M. Rubinov, Constructive Nonsmooth Analysis. Peter Lang, Frankfurt am Main (1995). Zbl0887.49014MR1325923
12. [12] V.F. Demyanov, A.M. Bagirov and A.M. Rubinov, A method of truncated codifferential with application to some problems of cluster analysis. J. Glob. Optim.23 (2002) 63–80. Zbl1034.49009MR1884966
13. [13] M.V. Dolgopolik, Codifferential Calculus in Normed Spaces. J. Math. Sci.173 (2011) 441–462. Zbl1290.46033MR2839880
14. [14] A.D. Ioffe, Euler–Lagrange and Hamiltonian formalism in dynamic optimization. Trans. Amer. Math. Soc.349 (1997) 2871–2900. Zbl0876.49024MR1389779
15. [15] A.D. Ioffe and R.T. Rockafellar, The Euler and Weierstrass conditions for nonsmooth variational problems. Calc. Var. Partial Differ. Equ.4 (1996) 59–87. Zbl0838.49015MR1379193
16. [16] A.D. Ioffe and V.M. Tihomirov, Theory of Extremal Problems. North-Holland, Amsterdam (1979). Zbl0407.90051MR528295
17. [17] P.D. Loewen and R.T. Rockafellar, New Necessary Conditions for the Generalized Problem of Bolza. SIAM J. Control Optim.34 (1996) 1496–1511. Zbl0871.49023MR1404843
18. [18] 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. Zbl0761.49003MR1184644
19. [19] B. Mordukhovich, Discrete approximations and refined Euler–Lagrange conditions for nonconvex differential inclusions. SIAM J. Control Optim.33 (1995) 882–915. Zbl0844.49017MR1327242
20. [20] D. Pallaschke and R. Urbański, Reduction of quasidifferentials and minimal representations. Math. Program.66 (1994) 161–180. Zbl0824.49016MR1297060
21. [21] R.T. Rockafellar, Conjugate convex functions in optimal control and the calculus of variations. J. Math. Anal. Appl.32 (1970) 174–222. Zbl0218.49004MR266020
22. [22] R.T. Rockafellar, Generalized Hamiltonian equations for convex problems of Lagrange. Pacific. J. Math.33 (1970) 411–428. Zbl0199.43002MR276853
23. [23] R.T. Rockafellar, Existence and duality theorems for convex problems of Bolza. Trans. Amer. Math. Soc.159 (1971) 1–40. Zbl0255.49007MR282283
24. [24] S. Scholtes, Minimal pairs of convex bodies in two dimensions. Mathematika39 (1992) 267–273. Zbl0759.52004MR1203283
25. [25] R. Vinter and H. Zheng, The Extended Euler–Lagrange Condition for Nonconvex Variation Problems. SIAM J. Control Optim.35 (1997) 56–77. Zbl0870.49014MR1430283
26. [26] R. Vinter, Optimal Control. Birkhauser, Boston (2000). Zbl1215.49002MR1756410
27. [27] K. Yosida, Functional Analysis. Springer-Verlag, New York (1980). Zbl0435.46002MR617913