Introduction to Jan C. Willems’, “Dissipative Dynamical Systems, Part II: Linear Systems With Quadratic Supply Rates”

The two-part article, “Dissipative Dynamical Systems,” by Jan C. Willems, was submitted in January 1972 to the
Archive for Rational Mechanics and Analysis
and published in the same year (vol. 45, Part I, pp. 321–351, Part II: pp. 352–393). While Part I develops the theory of dissipative systems for general input–state–output systems, Part II confines itself to
linear systems
. Moreover, most of Part II is about dissipativity with respect to the supply rate
$y{\hspace{0.06em}}^{\top}u,$
that is, linear
passive
systems. The main questions addressed in Part II concern the analysis of the set of storage functions of linear passive systems (with the existence of a minimal and maximal element already established in Part I) and the compatibility of storage functions with other external and internal symmetry properties of the system, such as reciprocity and reversibility. (See “The Articles of This Special Issue.”) This links the article closely to the realization theory of
physical
linear systems, in particular, the network synthesis of electrical circuits, the Onsager–Casimir relation of linear thermodynamics, and the input–output analysis of linear viscoelastic systems.