Global asymptotic stability of nonlinear cascade systems (Q1861755)

Consider the nonlinear cascade system \((*)\) \(dx/dt = f(x,y)\), \(dy/dt = g(y),\) where \(f:\mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^n\) and \(g: \mathbb{R}^m \rightarrow \mathbb{R}^m\) are assumed to be \(C^1\)-functions and to satisfy \(f(0,0)=0\), \(g(0)=0.\) The author gives a short proof of a known result on the global asymptotic stability of \((*)\) and provides a similar result for a discretre-time cascade system.