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Transactions of the American Mathematical Society

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Turnpike property for extremals
of variational problems
with vector-valued functions


Author: A. J. Zaslavski
Journal: Trans. Amer. Math. Soc. 351 (1999), 211-231
MSC (1991): Primary 49J99, 58F99
DOI: https://doi.org/10.1090/S0002-9947-99-02132-7
MathSciNet review: 1458340
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Abstract: In this paper we study the structure of extremals $\nu\colon[0,T]\to R^n$ of variational problems with large enough $T$, fixed end points and an integrand $f$ from a complete metric space of functions. We will establish the turnpike property for a generic integrand $f$. Namely, we will show that for a generic integrand $f$, any small $\varepsilon>0$ and an extremal $\nu\colon[0,T]\to R^n$ of the variational problem with large enough $T$, fixed end points and the integrand $f$, for each $\tau\in[L_1, T-L_1]$ the set $\{\nu(t)\colon t\in[\tau,\tau+L_2]\}$ is equal to a set $H(f)$ up to $\varepsilon$ in the Hausdorff metric. Here $H(f)\subset R^n$ is a compact set depending only on the integrand $f$ and $L_1>L_2>0$ are constants which depend only on $\varepsilon$ and $|\nu(0)|$, $|\nu(T)|$.


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  • 1. Z. Artstein and A. Leizarowitz, Tracking periodic signals with overtaking criterion, IEEE Trans. on Autom. Control AC 30 (1985), 1122-1126.
  • 2. J. P. Aubin and I. Ekeland, Applied nonlinear analysis, Wiley Interscience, New York, 1984. MR 87a:58002
  • 3. J. M. Ball and V. J. Mizel, One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation, Arch. Rational Mech. Anal. 90 (1985), 325-388. MR 86k:49002
  • 4. W. A. Brock and A. Haurie, On existence of overtaking optimal trajectories over an infinite horizon, Math. Oper. Res. 1 (1976), 337-346. MR 56:14975
  • 5. D. A. Carlson, The existence of catching-up optimal solutions for a class of infinite horizon optimal control problems with time delay, SIAM Journal on Control and Optimization 28 (1990), 402-422. MR 91b:49008
  • 6. D. A. Carlson, A. Haurie and A. Jabrane, Existence of overtaking optimal solutions to infinite dimensional control problems on unbounded time intervals, SIAM Journal on Control and Optimization 25 (1987), 1517-1541. MR 88j:90063
  • 7. D. A. Carlson, A. Haurie and A. Leizarowitz, Infinite horizon optimal control, Springer-Verlag, Berlin, 1991.
  • 8. B. D. Coleman, M. Marcus and V. J. Mizel, On the thermodynamics of periodic phases, Arch. Rational Mech. Anal. 117 (1992), 321-347. MR 93d:73008
  • 9. J. L. Kelley, General topology, D. Van Nostrand Co., 1955. MR 51:6681
  • 10. A. Leizarowitz, Existence of overtaking optimal trajectories for problems with convex integrands, Math. Oper. Res. 10 (1985), 450-461. MR 87f:49002
  • 11. A. Leizarowitz, Infinite horizon autonomous systems with unbounded cost, Appl. Math. and Opt. 13 (1985), 19-43. MR 86g:49002
  • 12. A. Leizarowitz, Optimal trajectories on infinite horizon deterministic control systems, Appl. Math. and Opt. 19 (1989), 11-32. MR 89m:49049
  • 13. A. Leizarowitz and V. J. Mizel, One dimensional infinite-horizon variational problems arising in continuum mechanics, Arch. Rational Mech. Anal. 106 (1989), 161-194. MR 90b:49007
  • 14. V. L. Makarov and A. M. Rubinov, Mathematical theory of economic dynamics and equilibria, Nauka, Moscow, 1973; English transl., Springer-Verlag, 1977. MR 55:11973
  • 15. C. B. Morrey, Multiple integrals in the calculus of variations, Springer, Berlin Heidelberg New York, 1966. MR 34:2380
  • 16. A. M. Rubinov, Economic dynamics, in Itogy Nauki. Sovremennye problemy mat. 19 (1982) VINITI Moscow, 59-110; English transl., J. Soviet Math. 26 (1984), 4. MR 83k:90034
  • 17. A. J. Zaslavski, Optimal programs on infinite horizon I, SIAM Journal on Control and Optimization 33 (1995), 1643-1660. MR 96i:49047
  • 18. A. J. Zaslavski, Optimal programs on infinite horizon II, SIAM Journal on Control and Optimization 33 (1995), 1661-1686. MR 96i:49047
  • 19. A. J. Zaslavski, The existence and structure of extremals for a class of second order infinite horizon variational problems, Journal of Mathematical Analysis and Applications 194 (1995), 660-696. MR 96h:49005
  • 20. A. J. Zaslavski, Dynamic properties of optimal solutions of variational problems, Nonlinear Analysis: Theory, Methods and Applications 27 (1996), 895-931. MR 97h:49022
  • 21. A. J. Zaslavski, Structure of extremals for one-dimensional variational problems arising in continuum mechanics, Journal of Mathematical Analysis and Applications 198 (1996), 893-921. MR 97a:49006

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Additional Information

A. J. Zaslavski
Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000, Israel
Email: ajzasl@techunix.technion.ac.il

DOI: https://doi.org/10.1090/S0002-9947-99-02132-7
Keywords: Good function, turnpike property, representation formula, minimal long-run average cost growth rate
Received by editor(s): September 29, 1995
Received by editor(s) in revised form: November 18, 1996
Article copyright: © Copyright 1999 American Mathematical Society

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