Stability theory for functional-differential equations

Abstract:

We consider a system of functional differential equations $x’ (t) = \mathcal {F} (t, x( \cdot ))$, together with a Liapunov functional $\mathcal {V} (t, x( \cdot ))$ with $\mathcal {V}’ \leqslant 0$. Most classical results require that $\mathcal {F}$ be bounded for $x( \cdot )$ bounded and that $\mathcal {F}$ depend on $x(s)$ only for $t - \alpha (t) \leqslant s \leqslant t$ where $\alpha$ is a bounded function in order to obtain stability properties. We show that if there is a function $H(t, x)$ whose derivative along $x’ (t) = \mathcal {F} (t, x( \cdot ))$ is bounded above, then those requirements can be eliminated. The derivative of H may take both positive and negative values. This extends the classical theorem on uniform asymptotic stability, gives new results on asymptotic stability for unbounded delays and unbounded $\mathcal {F}$, and it improves the standard results on the location of limit sets for ordinary differential equations.