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

Abstract

The two-part article, “Dissipative Dynamical Systems,” by Jan C. Willems, was submitted in January 1972 to the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Archive for Rational Mechanics and Analysis</i> 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">linear systems</i> . Moreover, most of Part II is about dissipativity with respect to the supply rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$y{\hspace{0.06em}}^{\top}u,$</tex-math></inline-formula> that is, linear <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">passive</i> 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">physical</i> 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.