Conditions for the existence of continuous storage functions for nonlinear dissipative systems

The problem of existence of continuous storage functions for dissipative nonlinear systems is considered. It is shown that, if a nonlinear system is dissipative in the state x_*, then, under certain assumptions, a continuous storage function can be constructed on a set of points accessible from x_* by concatenation of a finite number of forward and backward motions of the system. Most of these assumptions are weaker than certain controllability-type properties and can be checked using similar tests.