Math 2.0

Ordinary differential equations Stability Theorem

Lyapunov's stability theorem

Builds on Lyapunov functionLyapunov stabilityAsymptotic stability

Statement

Let UU be an open subset of ℝn\mathbb{R}^n, let f:U→ℝnf : U \to \mathbb{R}^n be locally Lipschitz, and let x∗∈Ux^* \in U be an equilibrium point of x′=f(x)x' = f(x). Let W⊆UW \subseteq U be an open neighbourhood of x∗x^* and let V:W→ℝV : W \to \mathbb{R} be a Lyapunov function for x∗x^*. (1) Then x∗x^* is Lyapunov stable. (2) If VV is a strict Lyapunov function, then x∗x^* is asymptotically stable. (3) If W=U=ℝnW = U = \mathbb{R}^n, VV is strict, and V(x)→∞V(x) \to \infty as ∥x∥→∞\|x\| \to \infty, then x∗x^* is globally asymptotically stable.

Remarks

Proof idea for (1): choose ε\varepsilon so small that the sphere SS of radius ε\varepsilon around x∗x^* lies in WW. Let m>0m > 0 be the minimum of VV on SS. A solution that starts where V<mV < m keeps V<mV < m, so it cannot reach SS. For the damped harmonic oscillator x′=yx' = y, y′=−x−cyy' = -x - c y with c>0c > 0, the energy V=12(x2+y2)V = \frac{1}{2}(x^2 + y^2) has V˙=−cy2≤0\dot{V} = -c y^2 \le 0. This gives stability, and LaSalle's invariance principle gives asymptotic stability. The method needs no solution formula. The difficulty is to find a suitable VV.

History

The method is named after Aleksandr Lyapunov, who introduced it in his study of the stability of motion.

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

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Differential equation, concept, no evidence levelDifferential equationconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionGrönwall's inequality, statement, no evidence levelGrönwall's inequalitystatementSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionAutonomous system, definition, no evidence levelAutonomous systemdefinitionInitial value problem, definition, no evidence levelInitial value problemdefinitionEquilibrium point, definition, no evidence levelEquilibrium pointdefinitionUniqueness of solutions for Lipschitz vector fields, statement, no evidence levelUniqueness of solutio…statementLyapunov function, definition, no evidence levelLyapunov functiondefinitionMaximal solution, definition, no evidence levelMaximal solutiondefinitionFlow of a vector field, definition, no evidence levelFlow of a vector fielddefinitionLyapunov stability, definition, no evidence levelLyapunov stabilitydefinitionAsymptotic stability, definition, no evidence levelAsymptotic stabilitydefinitionLyapunov's stability theorem, statement, no evidence levelLyapunov's stability …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Differential equation concept
  2. Function concept
  3. Interval concept
  4. Ordinary differential equation definition
  5. Lipschitz condition definition
  6. Grönwall's inequality statement
  7. Solution of an ordinary differential equation definition
  8. Autonomous system definition
  9. Initial value problem definition
  10. Equilibrium point definition
  11. Uniqueness of solutions for Lipschitz vector fields statement
  12. Lyapunov function definition
  13. Maximal solution definition
  14. Flow of a vector field definition
  15. Lyapunov stability definition
  16. Asymptotic stability definition
  17. Lyapunov's stability theorem statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Lyapunov_function, Basic Lyapunov theorems for autonomous systems
About this page
Tag0J6A752DTZ
URIm2://0J6A752DTZ
Aliaseslyapunov-direct-method
Kindstatement, theorem
WritingW0 stub
MSC34D20
StatusThis copy of the site has no build data. Run m2 status 0J6A752DTZ.
UsesLyapunov function, Lyapunov stability, Asymptotic stability

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