The authors consider the system \(\dot x= f(x, u)\), \(f: \mathbb{R}^n\times \mathbb{R}^m\to \mathbb{R}^n\), locally Lipschitz with \(f(0, 0)= 0\). The system is ISS if there exists a positive definite \(V(x)\), \(V(0)= 0\), radially bounded \(\mathbb{R}^n\to \mathbb{R}\) (a storage function) and two functions (in \(K_\infty\)) \(\alpha\) and \(\gamma\) such that \(\dot V= (\nabla V\cdot f)\leq \gamma(|u|)- \alpha(|x|)\). A combination of the \(\gamma\) and \(\alpha\) functions characterizes the “input to gain” for the system. The authors prove some relations for positive definiteness of the storage function and the existence of \(K_\infty\) functions assuring that dissipation estimates for some composite systems are available.