# Lyapunov functions

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

• For some continuous function $f:{\mathbb R}^n\rightarrow {\mathbb R}^n$, we suppose $x(t)$ obeys the differential equation
• A Lyapunov function is a continuously differentiable function $L:{\mathbb R}^n\rightarrow {\mathbb R}$ with unique minimum at $x^*$ such that

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

• We add the additional assumption that $\{x: L(x)\leq l\}$ is a compact set for every $l\in{\mathbb R}$.

Thrm [Lya:ConvThrm] If a Lyapunov exists for differential equation then $L(x(t))\searrow L(x^*)$ as $t\rightarrow\infty$ and

Proof: Firstly,

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

So $L(x(t))$ is decreasing. Suppose it decreases to $\tilde{L}$. By the Fundamental Theorem of Calculus

Thus we can take a sequence of times $\{s_k\}_{k=1}^\infty$ such that $\frac{d L(x(s_k))}{dt}\rightarrow 0$ as $s_k\rightarrow\infty$. As $\{ x : L(x)< L(x(0))\}$ is compact, we can take a subsequence of times $\{t_k\} _{k=1}^\infty\subset \{s_k\}_{k=1}^\infty$, $t_k\rightarrow\infty$ such that $\{x(t_k)\}_{k=0}^\infty$ converges. Suppose it converges to $\tilde{x}$. By continuity,

Thus by definition $\tilde{x}=x^*$. Thus $\lim_{t\rightarrow\infty} L(x(t))=L(x^*)$ and thus by continuity of $L$ at $x^*$ we must have $x(t)\rightarrow x^*$.

• One can check this proof follows more-or-less unchanged if $x^*$, the minimum of $L$, is not unique.

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

If we further assume that $f$ and $L$ satisfy the conditions

1. $f(x)\cdot \nabla L(x) \leq -\gamma L(x)$ for some $\gamma>0$.
2. $\exists$ $\alpha , \eta>0$ such that $\alpha || x^*-x||^\eta \leq L(x)-L(x^*)$.
3. $L(x^*)=0$.

then there exists a constants $\kappa,K>0$ such that for all $t\in{\mathbb R}_+$

So long as $x(t) \neq x^*$, $L(x(t))>0$, thus dividing by $L(x(t))$ and integrating gives

Rearrganging gives

This gives exponential convergence in $L(x(t))$ and quick application of the bound in the 2nd assumption gives

• We can assume the 2nd assumption only holds on a ball arround $x^*$. We have convergence from Theorem [Lya:ConvThrm], so when $x(t)$ 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 $L$ is positive definite at $x^*$.