On stability analysis of systems featuring a multiplicative combination of nonlinear and linear time-invariant feedback

We show that a set of sufficient conditions for the finite-gain stability of certain single-input-single-output (SISO) systems featuring a multiplicative combination of memoryless nonlinear feedback and linear time-invariant feedback, as encountered in some systems biology prototype applications, can be obtained by building on the Rantzer multipliers. These results are obtained by observing that for all finite energy signals x, the time-truncated inner product 〈x, H(x)N(x)〉_T is positive for all T >; 0 if N belongs to a subclass of positive memoryless monotone nonlinearities and if, in addition, H is a Rantzer multiplier.