Some applications of spectral-radius concept to nonlinear feedback stability

Several results are derived concerning the input-output stability of nonlinear time-varying feedback systems. These results all share the common feature that they are based on an application of the concept of the spectral radius of a bounded linear operator. Section I contains the preliminary notions, including that of the spectral radius. In Section II, bounds are obtained for the spectral radius of a Volterra integral operator, and these bounds are used to obtain sufficient conditions for the existence of an inverse operator for a type of nonlinear operator. This technique is applied to obtain stability regions for the Mathieu-Hill equation, in order to illustrate the fact that the method proposed here yields less conservative stability bounds than those obtained by standard contraction methods. In Section III, several results are proved regarding the existence of inverse operators for linear operators. In Section IV, the results of Section III are applied to the
L_{p}
-stability problem for linear timevarying systems. It is shown that, under certain conditions, the well-known circle criterion implies
L_{p}
-stability for all
p
, rather than just
L_{2}
-stability.