Dissipative energy functionals of passive linear time-varying systems
The concept of dissipativity plays a crucial role in the analysis of control systems. Dissipative energy functionals, also known as Hamiltonians, storage functions, or Lyapunov functions, depending on the setting, are extremely valuable to analyze and control the behavior of dynamical systems, but in general circumstances they are very difficult to compute, and not fully understood. In this paper we consider passive linear time-varying (LTV) systems, under very mild regularity assumptions, and their associated storage functions, as a necessary step to analyze general nonlinear systems. We demonstrate that every passive LTV system must have at least one time-varying positive semidefinite quadratic storage function, greatly reducing our search scope. Now focusing on quadratic storage functions, we analyze in detail their necessary regularity, which is lesser than continuous. Moreover, we prove that the rank of quadratic storage functions is nonincreasing in time, allowing us to introduce a novel null space decomposition, under much weaker assumptions than the one needed for general matrix functions. Additionally, we show a necessary kernel condition for the quadratic storage function, allowing us to reduce our search scope even further.