Dissipative Dynamical Systems

Dissipative systems provide a strong link between physics, system theory, and control engineering. Dissipativity is first explained in the classical setting of input/state/output systems. In the context of linear systems with quadratic supply rates, the construction of a storage leads to a linear matrix inequality (LMI). It is in this context that LMI's first emerged in the field. Next, we phrase dissipativity in the setting of behavioral systems, and present the construction of two canonical storages, the available storage and the required supply. This leads to a new notion of dissipativity, purely in terms of boundedness of the free supply that can be extracted from a system. The storage is then introduced as a latent variable associated with the supply rate as the manifest variable. The equivalence of dissipativity with the existence of a non-negative storage is proven. Finally, we deal with supply rates that are given as quadratic differential forms and state several results that relate the existence of a (non-negative) storage to the two-variable polynomial matrix that defines the quadratic differential form. In the ECC presentation, we mainly discuss distributed dissipative systems described by constant coefficient linear PDE's. In this setting, the construction of storage functions leads to Hilbert's 17-th problem on the representation of non-negative polynomials as a sum of squares.