I want to show that a set of differential equations with a fixed delay and with finite discontinuities converges to some equilibrium. I've simulated it and used a rather ugly $\epsilon,\delta$ kind of proof to show where all the equilibria are, but I'd like a simpler proof, and I think a Lyapunov function would be good.
Is there a way to construct a Lyapunon function for a differential equation with a delay term?
Indeed, it is possible sometimes to construct Lyapunov functions for a delay equation. Note that the complication is caused by the discontinuities, not by the delay (although I realize that the opposite would be expected).
The prescription is similar to the usual one in terms of derivatives. I regret that I cannot afford the time to wrote down the details, but I suggest the following: rewrite the usual conditions for a Lyapunov function without using the derivative, that is, $\dot V(x)=\nabla V(x)\cdot f(x)$ for a Lyapunov function $V$ associated to a differential equation $x'=f(x)$, in terms of the flow. This means the following: for example, if the equation indices a flow $\varphi_t$, the condition $\dot V(x)\le0$ is equivalent to $V(\varphi_t(x))\le V(x)$. With this procedure you can treat the discontinuities as if they were not there. In other words, you should use the condition $V(\varphi_t(x))\le V(x)$ instead of $\dot V(x)\le0$, since it does not require derivatives!
As for the stable behavior of the delay equation, you will have to proceed in similar ways as in the classical case. To find out a Lypunov function is sometimes more art than mathematics, and it is the same with delay equations. That depends on the structure of your equation, its properties, etc, so without more information nothing more can be said.