Some Stability Results Related to Those of V. M. Popov

To a considerable extent, Ref. 1 is a summary of certain results of a recent study by this writer of the input-output properties of a large class of time-varying nonlinear systems. The properties of a vector nonlinear Volterra integral equation of the second kind that frequently arises in the study of physical systems are considered in detail,* and some conditions are presented for the norm-boundedness of solutions of a functional equation of similar type defined on an abstract space. Much of the material presented in Ref. 1 is drawn from Refs. 2 and 3. In Ref. 1, some techniques other than those of Refs. 2 and 3 are described for obtaining sufficient conditions for the £ 2 -boundedness and JB^-boundedness of solutions of functional equations. In this paper, these techniques are developed further and are used to prove some stability results, related to those of V. M. Popov, 8 for large classes of feedback systems and electrical networks that contain subsystems which are not necessarily representable in terms of ordinary differential equations, f I I I . THE FEEDBACK SYSTEM AND THE MAIN R E S U L T S