###### tags: `one-offs` `lyapunov functions`

# Contractivity from Eventual Exponential Convergence

**Overview**: In this note, I describe some thoughts on Lyapunov functions, as relates to stability of dynamical systems, and how one might improve an a priori estimate of 'eventual exponential convergence' into a refined estimate of 'uniform-in-time contractivity'.

## Construction

There is a construction about which I'm a bit curious: suppose that you have a moving particle $x$ which eventually reaches equilibrium at an exponential rate, in the sense that there is some nice function $V$ and a positive, finite function $M$ such that

\begin{align}
t\geqslant0,\quad V\left(x\left(t\right)\right)\leqslant M\left(x\right)\cdot\exp\left(-c\cdot t\right),
\end{align}

where the presence of $M$ allows for the possibility that $V$ might not decrease monotonically, initially or otherwise.

Moving forward, you can now define

\begin{align}
W\left(x\right):=\sup\left\{ \exp\left(c\cdot t\right)\cdot V\left(x\left(t\right)\right):t\geqslant0\right\} ,
\end{align}

and check that $V\leqslant W\leqslant M$; in particular, $W$ is certainly finite.

With a little bit of extra work, one then obtains that

\begin{align}
\forall t \geqslant 0, \quad W\left(x\left(t\right)\right)\leqslant W\left(x\right)\cdot\exp\left(-c\cdot t\right)
\end{align}

and so by a reverse version of Grönwall's inequality, one obtains that that $W$ will be uniformly exponentially decreasing for all time, i.e. that

\begin{align}
\langle \nabla W(x), \dot{x} \rangle \leqslant - c \cdot W(x).
\end{align}

## Recap

The idea is then that if $V$ is eventually exponentially decreasing, then one can always find a $W$ which majorises $V$, and which is *uniformly* exponentially decreasing, which is somehow one of the nicest 'certificates' of this type of decay.

My curiosity essentially concerns whether this sort of construction has a particular name. That is, given a Lyapunov function with 'eventual exponential decay', identify a modified Lyapunov function with 'uniform exponential decay'. It seems like this construction shouldn't be new, and so I am curious about what people have worked out about it already.