Zames-Falb multipliers for MIMO nonlinearities

In their celebrated 1968 paper on nonlinear stability, Zames and Falb deter mined a class of multipliers that preserve positivity of monotone SISO nonl inearities. They conjectured that their results might also hold for increme ntally positive, norm-bounded MIMO nonlinearities. In this note, we demonst rate that their conjecture regarding MIMO nonlinearities holds true only if a further restriction;is applied. Specifically, we show that it suffices e ither to restrict the nonlinearity to be the gradient of a convex real-valu ed function or to restrict the multiplier to be a real-valued function of f requency. Copyright (C) 2000 John Wiley & Sons, Ltd.