Application of Lyapunov's second method to the analysis of general second order bang-bang control systems

Because of the complications involved in synthesizing time optimal switching functions, several approximate or quasi-time optimal switching functions have been considered. Of importance in selecting such an approximate switching curve is whether or not the corresponding forcing function will reduce the plant's initial states to the origin, thus giving asymptotic stability. This papers by using the Second Method of Lyapunov, develops a set of sufficient conditions which, if met by the switching curve and the plant, will insure the existence of asymptotic stability in the large. The plant considered is defined by the general second-order expression
\ddot{c} + g(c)h(\dot{c})\dot{c} + p(c) = m
.