  The theory of dissipativity has been primarily developed for controllable
systems/behaviors. For various reasons, in the context of uncontrollable
systems/behaviors, a more appropriate definition of dissipativity is in terms
of the dissipation inequality, namely the {\em existence} of a storage
function. A storage function is a function such that along every system
trajectory, the rate of increase of the storage function is at most the power
supplied. While the power supplied is always expressed in terms of only the
external variables, whether or not the storage function should be allowed to
depend on unobservable/hidden variables also has various consequences on the
notion of dissipativity: this paper thoroughly investigates the key aspects of
both cases, and also proposes another intuitive definition of dissipativity.
  We first assume that the storage function can be expressed in terms of the
external variables and their derivatives only and prove our first main result
that, assuming the uncontrollable poles are unmixed, i.e. no pair of
uncontrollable poles add to zero, and assuming a strictness of dissipativity at
the infinity frequency, the dissipativities of a system and its controllable
part are equivalent. We also show that the storage function in this case is a
static state function.
  We then investigate the utility of unobservable/hidden variables in the
definition of storage function: we prove that lossless autonomous behaviors
require storage function to be unobservable from external variables. We next
propose another intuitive definition: a behavior is called dissipative if it
can be embedded in a controllable dissipative {\em super-behavior}. We show
that this definition imposes a constraint on the number of inputs and thus
explains unintuitive examples from the literature in the context of
lossless/orthogonal behaviors.
