Systems & Control Letters 46 (2002) 85–90 www.elsevier.com/locate/sysconle On the existence of a continuous storage function for dissipative systems I.G. Polushin ∗;1 , H.J. Marquez Deparment of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada T6G2G7 Received 15 March 2001; received in revised form 30 November 2001 Abstract Conditions for nonlocal existence of a continuous storage function for nonlinear dissipative system are presented. More precisely, it is shown that under the local w-uniform reachability assumption at one point x, the required supply function is continuous on the set of points reachable from x. Conditions for the local w-uniform reachability based on the local controllability properties of the system are provided. c  2002 Elsevier Science B.V. All rights reserved. Keywords: Dissipative systems; Storage functions; Nonlinear systems; Local w-uniform reachability 1. Introduction In recent years the theory of dissipative systems has played an essential role in nonlinear control (see [15,7,10] and references therein). One of the main results of this theory is the equivalence be- tween the dissipativity property of a system and the existence of a storage function. In general, storage functions are discontinuous, and not many results concerning the existence of more regular (continuous, smooth, etc.) storage functions have been obtained. In particular, James [4] showed that for every storage function its lower semicontinuous envelope is also a storage function. Hill and Moylan [3] proved that un- der the assumption of “local w-uniform reachability at * Corresponding author. Tel.: +1-780-492-3334; fax: +1-780-492-1811. E-mail addresses: polushin@ee.ualberta.ca (I.G. Polushin), marquez@ee.ualberta.ca (H.J. Marquez). 1 On leave from CCS Laboratory, Institute for Problems of Mechanical Engineering, Russian Academy of Sciences. every state”, any storage function is continuous. Fur- ther, in the case in which the supply rate corresponds to the L 2 -gain of the system, using more advanced techniques van der Schaft [14] showed that usually a smooth storage function exists at least locally. The last result is important in nonlinear H ∞ control problems, since the control law which solves the problem depends on @V=@x. Also, a number of results concerned with the existence of smooth storage func- tions were obtained by Sontag and coauthors [11,5,12] for special types of supply rates corresponding to the input-to-state stability and related properties. In this paper we give sucient conditions for non- local existence of a continuous storage function for dissipative system with an arbitrary supply rate. More explicitly, we improve the result of Hill and Moylan [3] in the following aspect: instead of local w-uniform reachability at every point we assume that this prop- erty holds only at one single point x ∗ . Thus, we re- place the global assumption by a local one. We show that in this case the set R(x ∗ ) of the points reachable from x ∗ is open and connected and there exists a 0167-6911/02/$-see front matter c  2002 Elsevier Science B.V. All rights reserved. PII:S0167-6911(02)00108-1