no. 1
The limiting behavior of the value-function for variational problems arising in continuum mechanics
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1, pp. 57-72.

In this paper we study the limiting behavior of the value-function for one-dimensional second order variational problems arising in continuum mechanics. The study of this behavior is based on the relation between variational problems on bounded large intervals and a limiting problem on [0,∞).

DOI : 10.1016/j.anihpc.2009.07.005
Classification : 49J99
Mots clés : Good function, Infinite horizon, Minimal long-run average cost growth rate, Variational problem 
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     author = {Zaslavski, Alexander J.},
     title = {The limiting behavior of the value-function for variational problems arising in continuum mechanics},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Zaslavski, Alexander J. The limiting behavior of the value-function for variational problems arising in continuum mechanics. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1, pp. 57-72. doi : 10.1016/j.anihpc.2009.07.005. http://www.numdam.org/articles/10.1016/j.anihpc.2009.07.005/

[1] R.A. Adams, Sobolev Spaces, Academic Press, New York (1975) | Zbl

[2] S. Aubry, P.Y. Le Daeron, The discrete Frenkel–Kontorova model and its extensions I, Phys. D 8 (1983), 381-422 | Zbl

[3] J. Baumeister, A. Leitao, G.N. Silva, On the value function for nonautonomous optimal control problem with infinite horizon, Systems Control Lett. 56 (2007), 188-196 | Zbl

[4] L.D. Berkovitz, Lower semicontinuity of integral functionals, Trans. Amer. Math. Soc. 192 (1974), 51-57 | Zbl

[5] J. Blot, P. Cartigny, Optimality in infinite-horizon variational problems under sign conditions, J. Optim. Theory Appl. 106 (2000), 411-419 | Zbl

[6] J. Blot, P. Michel, The value-function of an infinite-horizon linear quadratic problem, Appl. Math. Lett. 16 (2003), 71-78 | Zbl

[7] B.D. Coleman, M. Marcus, V.J. Mizel, On the thermodynamics of periodic phases, Arch. Ration. Mech. Anal. 117 (1992), 321-347 | Zbl

[8] A. Leizarowitz, Tracking nonperiodic trajectories with the overtaking criterion, Appl. Math. Optim. 14 (1986), 155-171 | Zbl

[9] A. Leizarowitz, V.J. Mizel, One dimensional infinite horizon variational problems arising in continuum mechanics, Arch. Ration. Mech. Anal. 106 (1989), 161-194 | Zbl

[10] V.L. Makarov, A.M. Rubinov, Mathematical Theory of Economic Dynamics and Equilibria, Springer-Verlag, New York (1977) | Zbl

[11] M. Marcus, Universal properties of stable states of a free energy model with small parameters, Calc. Var. Partial Differential Equations 6 (1998), 123-142 | Zbl

[12] M. Marcus, A.J. Zaslavski, On a class of second order variational problems with constraints, Israel J. Math. 111 (1999), 1-28 | Zbl

[13] M. Marcus, A.J. Zaslavski, The structure of extremals of a class of second order variational problems, Ann. Inst. H. Poincaré Anal. Non Linéare 16 (1999), 593-629 | EuDML | Numdam | Zbl

[14] M. Marcus, A.J. Zaslavski, The structure and limiting behavior of locally optimal minimizers, Ann. Inst. H. Poincaré Anal. Non Linéare 19 (2002), 343-370 | EuDML | Numdam | Zbl

[15] B. Mordukhovich, Minimax design for a class of distributed parameter systems, Autom. Remote Control 50 (1990), 1333-1340 | Zbl

[16] B. Mordukhovich, I. Shvartsman, Optimization and feedback control of constrained parabolic systems under uncertain perturbations, Optimal Control, Stabilization and Nonsmooth Analysis, Lecture Notes Control Inform. Sci., Springer (2004), 121-132 | Zbl

[17] J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincaré Anal. Non Linéare 3 (1986), 229-272 | EuDML | Numdam | Zbl

[18] P.H. Rabinowitz, E. Stredulinsky, On some results of Moser and of Bangert, Ann. Inst. H. Poincaré Anal. Non Linéare 21 (2004), 673-688 | EuDML | Zbl

[19] P.H. Rabinowitz, E. Stredulinsky, On some results of Moser and of Bangert. II, Adv. Nonlinear Stud. 4 (2004), 377-396 | EuDML | Zbl

[20] A.J. Zaslavski, Ground states in Frenkel–Kontorova model, Math. USSR Izv. 29 (1987), 323-354 | Zbl

[21] A.J. Zaslavski, The existence of periodic minimal energy configurations for one-dimensional infinite horizon variational problems arising in continuum mechanics, J. Math. Anal. Appl. 194 (1995), 459-476 | Zbl

[22] A.J. Zaslavski, The existence and structure of extremals for a class of second order infinite horizon variational problems, J. Math. Anal. Appl. 194 (1995), 660-696 | Zbl

[23] A.J. Zaslavski, Structure of extremals for one-dimensional variational problems arising in continuum mechanics, J. Math. Anal. Appl. 198 (1996), 893-921 | Zbl

[24] A.J. Zaslavski, Existence and structure of optimal solutions of variational problems, Proceedings of the Special Session on Optimization and Nonlinear Analysis, Joint AMS-IMU Conference, Jerusalem, May 1995, Contemp. Math. vol. 204 (1997), 247-278 | Zbl

[25] A.J. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control, Springer, New York (2006) | Zbl

[26] A.J. Zaslavski, On a class of infinite horizon variational problems, Comm. Appl. Nonlinear Anal. 13 (2006), 51-57 | Zbl

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