  The dissipativity concept sits at the intersection of physics, systems theory
and control engineering, as a natural generalisation of passive systems that
dissipate energy. It relates properties of the external behavior of systems to
their internal state, and connects such wide-ranging subjects as optimal
control, algebraic Riccati equations, linear matrix inequalities, complex
functions, and spectral factorization. Its applications include the analysis
and design of interconnected systems (such as cyber-physical systems),
robustness and the absolute stability problem (the passivity, small-gain,
circle and Popov theorems and the theory of integral quadratic constraints),
and network synthesis (of electrical, mechanical and multi-physics systems). In
this article, we detail recent developments in the treatment of dissipativity
theory for systems that are not necessarily controllable and that need not lend
themselves naturally to an input-state-output perspective, drawing inspiration
from the behavioral theory of Jan Willems and collaborators. Such systems are
prevalent among physical systems, and we will illustrate the considered
concepts using simple electric circuit examples.
