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

## TL;DR

This paper establishes that negative upper-Lyapunov exponents characterize exponential stability of invariant sets in nonautonomous functional differential equations with state-dependent delays, linking stability to spectral properties.

## Contribution

It introduces a method to verify exponential stability via upper-Lyapunov exponents for nonautonomous FDEs with state-dependent delays, extending stability analysis tools.

## Key findings

- Negative upper-Lyapunov exponent implies exponential stability.
- Exponential stability of minimal sets is characterized by Lyapunov exponents.
- Existence of stable almost periodic solutions under stability conditions.

## 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.