Advertisement

Asymptotic Stability with a Discounted Criterion; Global and Local Analysis

  • D. A. Carlson
  • A. Haurie
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 290)

Abstract

In this chapter, the global asymptotic stability (G.A.S.) property of optimal controlled systems with an infinite time horizon will be further explored by considering the case where the criterion has the following form:
$$ {J_8}\left( {x,u} \right) = \int\limits_0^8 {{e^{ - pt}}{t_0}\left( {x\left( t \right),u\left( t \right)} \right)dt} $$
(5.1)
The term e -ρt is a discount factor and ρ is the discount rate. If ρ is positive and fo(x,u) is bounded then the integral converges when 8→∞. A positive discount rate is used when the future earnings have to be discounted due to an interest rate ρ. In that case the relative weighting given to a distant future makes it negligible, hence one may suppose that high positive values of ρ can have a destabilizing effect on the optimal trajectories.

Keywords

Stationary Point Discount Rate Hamiltonian System Optimal Trajectory Global Asymptotic Stability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. Cass and K. Shell, see Ref. 10 of Chapter 4.Google Scholar
  2. 2.
    W. Brock and J. Sheinkman, see Ref. 11 of Chapter 4.Google Scholar
  3. 3.
    T. Rockafellar, Saddle points of Hamiltonian Systems in Convex Lagrange Problems having a non-zero Discount Rate, Journal Economic Theory Vol. 12, pp. 71–113, 1976.CrossRefGoogle Scholar
  4. 4.
    M. Magill, On a General Economic Theor of Motion Springer Verlag, N.Y. 1970.CrossRefGoogle Scholar
  5. 5.
    M. Magill, Some New Results on the Local Stability of the Process of Capital Accumulation, Journa of Economic Theory Vol. 15, pp. 174–210, 1977.CrossRefGoogle Scholar
  6. 6.
    K. Shell, The Theory of Hamiltonian Dynamic Systems, in J. D. Groke ed. The Theor and Application of Differential Games D. Reidel 1975.Google Scholar
  7. 7.
    W. Brock and J. Sheinkman, see Ref. 12 of Chapter 4.Google Scholar
  8. 8.
    A. Bryson and Y. C. Ho, Applied Optimal Control Blaisdell 1969.Google Scholar
  9. 9.
    G. Leitmann, See Ref. 5 of Chapter 2.Google Scholar

Additional Reference

  1. 10.
    C. D. Feinstein and S. S. Oren, A “funnel” turnpike theorem for optimal growth problems with discounting, Journal of Economic Dynamics and Control Vol. 9, pp. 25–39, 1985.CrossRefGoogle Scholar
  2. 11.
    C. D. Feinstein and D. G. Luenberger, Analysis of the asymptotic behavior of optimal control trajectories: The implicit programming problem, SIAM Journal on Control and Optimization Vol. 19, pp. 561–585, 1981.CrossRefGoogle Scholar
  3. 12.
    H. Atsumi, Neoclassical growth and the efficient program of capital accumulation, Review of Economic Studies Vol. 30, pp. 127–136, 1963.Google Scholar
  4. 13.
    See Ref. 1, Chapter 4.Google Scholar
  5. 14.
    See Ref. 2, Chapter 4.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • D. A. Carlson
    • 1
  • A. Haurie
    • 2
    • 3
  1. 1.Department of MathematicsSouthern Illinois University at CarbondaleCarbondaleUSA
  2. 2.École des Hautes Études CommercialesMontréalCanada
  3. 3.École Polytechnique de MontréalMontréalCanada

Personalised recommendations