Lyapunov Stability

Lyapunov Stability is a stability criterion that described the behaviour of the solutions of differential equations near an equilibrium point.

• • …
• Counter-Example(s):
  • Linear Stability,
  • Orbital Stability,
  • von Neumann Stability Analysis,
  • Structural Stability.
• See: Input-to-state Stability, Stability Criterion, Asymptotic Analysis, Structural Stability, Lyapunov Function, Dynamical System, Asymptotic Stability, Hyperstability, Stability Radius, Perturbation Theory, LaSalle's Invariance Principle.

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References

2018

• (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Lyapunov_stability Retrieved:2018-10-14.
  • Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Lyapunov. In simple terms, if the solutions that start out near an equilibrium point [math]\displaystyle{ x_e }[/math] stay near [math]\displaystyle{ x_e }[/math] forever, then [math]\displaystyle{ x_e }[/math] is Lyapunov stable. More strongly, if [math]\displaystyle{ x_e }[/math] is Lyapunov stable and all solutions that start out near [math]\displaystyle{ x_e }[/math] converge to [math]\displaystyle{ x_e }[/math], then [math]\displaystyle{ x_e }[/math] is asymptotically stable. The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but "nearby" solutions to differential equations. Input-to-state stability (ISS) applies Lyapunov notions to systems with inputs.

1993

• (Murray et al., 1993) ⇒ R.M. Murray, Z. Li, and S.S. Sastry (1993). "Lyapunov Stability". In: "A Mathematical Introduction to Robotic Manipulation", CRC Press.
  • QUOTE: The equilibrium point [math]\displaystyle{ x^∗ = 0 }[/math] of (4.31) is stable (in the sense of Lyapunov) at [math]\displaystyle{ t = t_0 }[/math] if for any [math]\displaystyle{ \epsilon \gt 0 }[/math] there exists a [math]\displaystyle{ \delta(t_0,\epsilon) \gt 0 }[/math] such that

    [math]\displaystyle{ \parallel x(t_0)\parallel \lt δ \; \Longrightarrow \; \parallel x(t)\parallel \lt \epsilon,\; \forall_{t \ge t_0}\quad }[/math] (4.32)

     Lyapunov stability is a very mild requirement on equilibrium points. In particular, it does not require that trajectories starting close to the origin tend to the origin asymptotically.

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