Structure of approximate solutions of variational problems with extended-valued convex integrands

Zaslavski, Alexander J.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 4, pp. 872-894

In this work we study the structure of approximate solutions of autonomous variational problems with a lower semicontinuous strictly convex integrand $f$ : $R^n$$\times$$R^n$ $\to$ $R^1$$\cup$ $\left\{\infty \right\}$, where $R^n$ 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.

Classification : 49J99
Keywords: good function, infinite horizon, integrand, overtaking optimal function, turnpike property

[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). | Zbl | MR

[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. | Zbl | MR

[5] A. Leizarowitz, Infinite horizon autonomous systems with unbounded cost. Appl. Math. Opt. 13 (1985) 19-43. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR | Numdam

[8] L.W. Mckenzie Classical general equilibrium theory. The MIT press, Cambridge, Massachusetts, USA (2002). | Zbl | MR

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

[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. | Zbl | MR | Numdam

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

[12] R.T. Rockafellar, Convex analysis. Princeton University Press, Princeton, USA (1970). | Zbl | MR

[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. | Zbl | MR

[16] A.J. Zaslavski, Optimal programs on infinite horizon 2. SIAM J. Contr. Opt. 33 (1995) 1661-1686. | Zbl | MR

[17] A.J. Zaslavski, Turnpike properties in the calculus of variations and optimal control. Springer, New York (2006). | Zbl | MR

[18] A.J. Zaslavski, Structure of extremals of autonomous convex variational problems. Nonlinear Anal. Real World Appl. 8 (2007) 1186-1207. | MR

[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). | Zbl