On a stability theorem for nonlinear systems with slowly varying inputs

Recently, M. Kelemen (IEEE Trans. Automat. Contr., vol.AC-31, 766-768, Aug. 1986) presented a stability result that deals with the response of a nonlinear system to slowly varying input signals. In this work a proof is given, under a weaker hypothesis on the input signal, by constructing a Lyapunov function that is standard in the control literature. This proof may be more accessible than the other approaches. For completeness, brief proofs of some intermediate results that are regarded as well known are included. A simple example is given to show that this approach can yield explicit bounds in certain situations of interest