Stability Theory

  • Jan Willem Polderman
  • Jan C. Willems
Part of the Texts in Applied Mathematics book series (TAM, volume 26)


In this chapter we study the stability of dynamical systems. Stability is a very common issue in many areas of applied mathematics. Intuitively, stability implies that small causes produce small effects. There are several types of stability. In structural stability, one wants small parameter changes to have a similar small influence on the behavior of a system. In dynamic stability, which is the topic of this chapter, it is the effect of disturbances in the form of initial conditions on the solution of the dynamical equations that matters. Intuitively, an equilibrium point is said to be stable if trajectories that start close to it remain close to it. Dynamic stability is thus not in the first instance a property of a system, but of an equilibrium point. However, for linear systems we can, and will, view stability as a property of the system itself. In input/output stability small input disturbances should produce small output disturbances. Some of these concepts are intuitively illustrated by means of the following example.


Equilibrium Point Lyapunov Function Asymptotic Stability Imaginary Axis Polynomial Matrix 


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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Jan Willem Polderman
    • 1
  • Jan C. Willems
    • 2
  1. 1.Department of Applied MathematicsUniversity of TwenteEnschedeThe Netherlands
  2. 2.Mathematics InstituteUniversity of GroningenGroningenThe Netherlands

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