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)


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:
$$ {J_8}\left( {x,u} \right) = \int\limits_0^8 {{e^{ - pt}}{t_0}\left( {x\left( t \right),u\left( t \right)} \right)dt} $$
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.


Stationary Point Discount Rate Hamiltonian System Optimal Trajectory Global Asymptotic Stability 
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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

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