The bounded input-bounded output stability of «th-order systems is discussed in this work. In essence, one can demonstrate such stability when it is possible to construct a Liapunov function V whose total time derivative under the forcing function can be given by V ^ — r V + s J/1/2 where r and s are positive constants. This differential inequality implies a bounded response, and bounded input-bounded output stability is demonstrated via this method for an asymptotically stable constant coefficient system, an asymptotically stable linear periodic system, a forced non-linear system of the Lurie type that satisfies a Popov-type condition,, and a linear time- varying system satisfying a circle criterion. While similar results have been obtained (Kalman and Bertram 1960, Sandberg 1964 a, 1965 a, b. Zames 1965), the approach used in this work following that of Goldwyn et, al. (1966) has been extended to distributed parameter networks and systems (de Figueiredo and Chao 1969) and has been used in the study of global properties of non-Hurwitzian control systems (de Figueiredo and Dutertre 1970) © 1970 Taylor and Francis Group, LLC.