Partial Dissipativity Theory and Partial Stability of Feedback Interconnections for Nonlinear Dynamical Systems

In this paper, we develop partial dissipativity theory for nonlinear dynamical systems using basic input-output and state properties. Specifically, partial dissipativity is characterized using both an input-output as well as a state dissipation inequality involving a partial storage function that is nonnegative definite with respect to part of the system state. The results are then used to derive Kalman-Yakubovich-Popov conditions for characterizing necessary and sufficient conditions for partial dissipativity of nonlinear dynamical systems using continuously differentiable partial storage functions that are nonnegative definite with respect to part of the systems state. In addition, feedback interconnection partial stability results for nonlinear dynamical systems are developed thereby providing a generalization of the small gain and positivity theorems for guaranteeing partial stability of feedback systems.