# zbMATH — the first resource for mathematics

Structure of approximate solutions of variational problems with extended-valued convex integrands. (English) Zbl 1175.49002
Summary: We study the structure of approximate solutions of autonomous variational problems with a lower semicontinuous strictly convex integrand $$f:\mathbb R^n\times\mathbb R^n\to\mathbb R^1\cup\{\infty\}$$, where $$\mathbb R^n$$ is the $$n$$-dimensional Euclidean space. We obtain a full description of the structure of the approximate solutions which is independent of the length of the interval, for all sufficiently large intervals.

##### MSC:
 49J15 Existence theories for optimal control problems involving ordinary differential equations
Full Text:
##### References:
 [1] H. Atsumi, Neoclassical growth and the efficient program of capital accumulation. Rev. Econ. Studies 32 (1965) 127-136. [2] L. Cesari, Optimization - theory and applications. Springer-Verlag, New York (1983). Zbl0506.49001 MR688142 · Zbl 0506.49001 [3] D. Gale, On optimal development in a multi-sector economy. Rev. Econ. Studies 34 (1967) 1-18. [4] M. Giaquinta and E. Guisti, On the regularity of the minima of variational integrals. Acta Math. 148 (1982) 31-46. Zbl0494.49031 MR666107 · Zbl 0494.49031 · doi:10.1007/BF02392725 [5] A. Leizarowitz, Infinite horizon autonomous systems with unbounded cost. Appl. Math. Opt. 13 (1985) 19-43. Zbl0591.93039 MR778419 · Zbl 0591.93039 · doi:10.1007/BF01442197 [6] A. Leizarowitz and V.J. Mizel, One dimensional infinite horizon variational problems arising in continuum mechanics. Arch. Rational Mech. Anal. 106 (1989) 161-194. Zbl0672.73010 MR980757 · Zbl 0672.73010 · doi:10.1007/BF00251430 [7] M. Marcus and A.J. Zaslavski, The structure of extremals of a class of second order variational problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999) 593-629. Zbl0989.49003 MR1712568 · Zbl 0989.49003 · doi:10.1016/S0294-1449(99)80029-8 · numdam:AIHPC_1999__16_5_593_0 · eudml:78476 [8] L.W. McKenzieClassical general equilibrium theory. The MIT press, Cambridge, Massachusetts, USA (2002). Zbl1020.91002 MR1933283 · Zbl 1020.91002 [9] J. Moser, Minimal solutions of variational problems on a torus. Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986) 229-272. Zbl0609.49029 MR847308 · Zbl 0609.49029 · numdam:AIHPC_1986__3_3_229_0 · eudml:78113 [10] P.H. Rabinowitz and E. Stredulinsky, On some results of Moser and of Bangert. Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004) 673-688. Zbl1149.35341 MR2086754 · Zbl 1149.35341 · doi:10.1016/j.anihpc.2003.10.002 · numdam:AIHPC_2004__21_5_673_0 · eudml:78634 [11] P.H. Rabinowitz and E. Stredulinsky, On some results of Moser and of Bangert. II. Adv. Nonlinear Stud. 4 (2004) 377-396. Zblpre02149270 MR2100904 · Zbl 1229.35047 [12] R.T. Rockafellar, Convex analysis. Princeton University Press, Princeton, USA (1970). Zbl0193.18401 MR274683 · Zbl 0193.18401 [13] P.A. Samuelson, A catenary turnpike theorem involving consumption and the golden rule. Am. Econ. Rev. 55 (1965) 486-496. [14] C.C. von Weizsacker, Existence of optimal programs of accumulation for an infinite horizon. Rev. Econ. Studies 32 (1965) 85-104. [15] A.J. Zaslavski, Optimal programs on infinite horizon 1. SIAM J. Contr. Opt. 33 (1995) 1643-1660. Zbl0847.49021 MR1358089 · Zbl 0847.49021 · doi:10.1137/S036301299325726X [16] A.J. Zaslavski, Optimal programs on infinite horizon 2. SIAM J. Contr. Opt. 33 (1995) 1661-1686. Zbl0847.49022 MR1358089 · Zbl 0847.49022 · doi:10.1137/S0363012993257271 [17] A.J. Zaslavski, Turnpike properties in the calculus of variations and optimal control. Springer, New York (2006). Zbl1100.49003 MR2164615 · Zbl 1100.49003 · doi:10.1007/0-387-28154-1 [18] A.J. Zaslavski, Structure of extremals of autonomous convex variational problems. Nonlinear Anal. Real World Appl. 8 (2007) 1186-1207. Zblpre05168866 MR2331434 · Zbl 1186.49008 · doi:10.1016/j.nonrwa.2006.05.006 [19] A.J. Zaslavski, A turnpike result for a class of problems of the calculus of variations with extended-valued integrands. J. Convex Analysis (to appear). Zbl1162.49017 · Zbl 1162.49017 · www.heldermann.de
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.