Incremental positivity nonpreservation by stability multipliers

It is proved that stability multipliers, such as Zames-Falb (1968) multipliers, Popov (1961) multipliers and RL/RC multipliers, known to preserve the positivity of monotone memoryless nonlinearities, do not, in general, preserve their incremental positivity. Our result implies that stability results based on these multipliers should be interpreted with caution, since, without incremental positivity, the continuity of the closed-loop system's input-output relation may not be assured.