Exponential and global stability of nonlinear dynamical systems relative to initial time difference (Q628978)

The authors investigate the stability of solutions of a nonautonomous system \(\dot x = f(t, x)\) of nonlinear ordinary differential equations for \(f \in C({\mathbb R}_+ \times {\mathbb R}^n, {\mathbb R}^n)\) with respect to a difference in the initial times. To do so they develop a comparison principle for solutions with different starting times using Lyapunov functions \(V \in C({\mathbb R}_+ \times {\mathbb R}^n, {\mathbb R_+^m})\) that are locally Lipschitz in the space variable \(x\). Sufficient conditions for stability are derived. The theory is illustrated with an example for which \(x \in {\mathbb R}^2\).