  We revisit the classical dissipativity theorem of linear-quadratic theory in
a generalized framework where the quadratic storage is negative definite in a
p-dimensional subspace and positive definite in a complementary subspace. The
classical theory assumes p = 0 and provides an inter- connection theory for
stability analysis, i.e. convergence to a zero dimensional attractor. The
generalized theory is shown to provide an interconnection theory for
p-dominance analysis, i.e. convergence to a p-dimensional dominant subspace. In
turn, this property is the differential characterization of a generalized
contraction property for nonlinear systems. The proposed generalization opens a
novel avenue for the analysis of interconnected systems with low-dimensional
attractors.
