arXiv:1707.00148v1 [math.OC] 1 Jul 2017 The converse of the passivity and small-gain theorem for nonlinear input-output maps Sei Zhen Khong, Arjan van der Schaft * Version: March 2, 2017 Abstract We prove the following converse passivity theorem. Consider a sys- tem given by a causal nonlinear input-output map, mapping 0 onto 0. Then finite L 2 -gain of the feedback interconnection of this given sys- tem with an arbitrary passive system implies that the system is itself passive. The proof is based on the S-procedure lossless theorem due to Megretski & Treil, and is given in three slightly different versions. We discuss the importance of this result for the control of systems interacting with an unknown environment. Similarly, we provide in two versions a proof of the necessity of the small-gain condition for closed-loop stability of causal nonlinear input-output maps, extending the well-known necessity result in linear robust control. 1 Introduction The passivity and small-gain theorems are key to large parts of systems and control theory, see e.g. [13, 9, 12, 7, 11]. Both theorems provide a sta- bility ‘certificate’ when feedback interconnecting the given system with an unknown system which is either (in the small-gain setting) assumed to have an L 2 -gain smaller than the reciprocal of the L 2 -gain of the given system, or is passive like the given system. These theorems are valid (under appro- priate conditions) in a very general setting, from linear finite-dimensional systems to nonlinear and infinite-dimensional systems. This paper is con- cerned with converse versions of the passivity and small-gain theorem; that * Sei Zhen Kong is with the Institute for Mathematics and its Applications, The Uni- versity of Minnesota, Minneapolis, MN 55455, USA, szkhong@umn.edu, Arjan van der Schaft is with the Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Groningen, the Netherlands, a.j.van.der.schaft@rug.nl