Lyapunov functions are an extremely convenient device for proving that a dynamical system converges.

- For some continuous function , we suppose obeys the differential equation
- A Lyapunov function is a continuously differentiable function with unique minimum at such that

<img class=” aligncenter” title=”\label{Lya:Lya} f(x)\cdot \nabla L(x)

- We add the additional assumption that is a compact set for every .

**Thrm** [Lya:ConvThrm] If a Lyapunov exists for differential equation then as and

**Proof**: Firstly,

<img class=” aligncenter” title=”\frac{d L(x(t))}{dt} = f(x(t))\cdot \nabla L(x(t))

So is decreasing. Suppose it decreases to . By the Fundamental Theorem of Calculus

Thus we can take a sequence of times such that as . As is compact, we can take a subsequence of times , such that converges. Suppose it converges to . By continuity,

Thus by definition . Thus and thus by continuity of at we must have .

- One can check this proof follows more-or-less unchanged if , the minimum of , is not unique.

We now place some assumptions where we can make further comments about rates of convergence.

If we further assume that and satisfy the conditions

- for some .
- such that .
- .

then there exists a constants such that for all

So long as , , thus dividing by and integrating gives

Rearrganging gives

This gives exponential convergence in and quick application of the bound in the 2nd assumption gives

- We can assume the 2nd assumption only holds on a ball arround . We have convergence from Theorem [Lya:ConvThrm], so when is such that assumption 2 is satisfied we can then apply the same analysis for an exponential convergence rate. Ensuring the 2nd assumption locally is more easy to check, eg. check is positive definite at .

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