A dissipativity theorem for p-dominant systems
This work provides a novel theoretical framework for analyzing interconnected systems with low-dimensional attractors, which is relevant for control theory and nonlinear dynamics.
The paper extends the classical dissipativity theorem to p-dominant systems, enabling interconnection theory for convergence to p-dimensional attractors rather than zero-dimensional ones, thus providing a differential characterization of generalized contraction for nonlinear systems.
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.