Positivity and storage functions for quadratic differential forms

Differential equations and one-variable polynomial matrices play an essential role in describing dynamics of systems. When studying functions of the dynamical variables or specifying performance criteria in optimal control, we invariably encounter quadratic expressions in the variables and their derivatives. Two-variable polynomial matrices are the proper mathematical tool to express these quadratic functionals. We illustrate that dynamical equations expressed through one-variable polynomial matrices fit the functionals expressed through two-variable polynomial matrices