A linear dissipativity approach to incremental input-to-state stability for a class of positive Lur'e systems

Incremental stability properties are considered for certain systems of forced, nonlinear differential equations with a particular positivity structure. An incremental stability estimate is derived for pairs of input/state/output trajectories of the Lur'e systems under consideration, from which a number of consequences are obtained, including the incremental exponential input-to-state stability property and certain input-output stability concepts with linear gain. Incremental stability estimates provide a basis for an investigation into the response to convergent and (almost) periodic forcing terms, and is treated presently. Our results show that an incremental version of the real Aizerman conjecture is true for positive Lur'e systems when an incremental gain condition is imposed on the nonlinear term, as we describe. Our argumentation is underpinned by linear dissipativity theory – a property of positive linear control systems.