Exponential stability for nonautonomous functional differential equations with state-dependent delay

Abstract

The properties of stability of a compact set $K$ which is positively invariant for a semiflow $(\W\times W^{1,\infty}([-r,0],\mathbb{R}^n),\Pi,\mathbb{R}^+)$ determined by a family of nonautonomous FDEs with state-dependent delay taking values in $[0,r]$ are analyzed. The solutions of the variational equation through the orbits of $K$ induce linear skew-product semiflows on the bundles $K\times W^{1,\infty}([-r,0],\mathbb{R}^n)$ and $\mK\times C([-r,0],\mathbb{R}^n)$. The coincidence of the upper-Lyapunov exponents for both semiflows is checked, and it is a fundamental tool to prove that the strictly negative character of this upper-Lyapunov exponent is equivalent to the exponential stability of $K$ in $\W\times W^{1,\infty}([-r,0],\mathbb{R}^n)$ and also to the exponential stability of this compact set when the supremum norm is taken in $W^{1,\infty}([-r,0],\mathbb{R}^n)$. In particular, the existence of a uniformly exponentially stable solution of a uniformly almost periodic FDE ensures the existence of exponentially stable almost periodic solutions.