Stability Radii and Lyapunov Exponents

  • Fritz Colonius
  • Wolfgang Kliemann
Part of the Progress in Systems and Control Theory book series (PSCT, volume 6)


In the state space approach to stability of uncertain systems the concept of stability radius plays a central role. In this paper we use the classical concept of Lyapunov exponents, which describe the exponential growth behavior, in order to define a variety of stability and instability radii for families of linear systems = [A + u(t)]x, u(t) ∈ U ρ , ρ ≥ 0. Here {U ρ , ρ ≥ 0} can denote sets of real or complex matrices, and the perturbation u(t) can be deterministic or belong to different classes of stochastic processes. We analyze the stability radii and their relations, using Lyapunov exponents of associated bilinear control systems. The well developed theory of Lyapunov exponents for stochastic systems provides the background for the discussion of stability radii of systems excited by random noise. The example of the linear oscillator with uncertain restoring force is discussed in detail.


Lyapunov Exponent Large Deviation Principle Linear Oscillator Bilinear System Exponential Growth Rate 
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.


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Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Fritz Colonius
    • 1
  • Wolfgang Kliemann
    • 2
  1. 1.Institut für MathematikUniversität AugsburgAugsburgWest Germany
  2. 2.Department of MathematicsIowa State UniversityAmesUSA

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