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

Abstract

The properties of stability of compact set $\mathcal{K}$ which is positively invariant for a semiflow $(\Omega \times W^{1,\infty }([-r,0],{\mathbb{R}}^n),\Pi ,{\mathbb{R}}^{+})$ 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 $\mathcal{K}$ induce linear skew-product semiflows on the bundles $\mathcal{K}\times W^{1,\infty }([-r,0],{\mathbb{R}}^n)$ and $\mathcal{K}\times C([-r,0],{\mathbb{R}}^n)$. 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 $\mathcal{K}$ in $\Omega \times W^{1,\infty }([-r,0],{\mathbb{R}}^n)$ and also to the exponential stability of this minimal 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.