The notion of a dissipative dynamical system generalizes the idea of a Lyapunov function to 'open' dynamical systems. This concept has found applications in diverse areas of systems and control, for example, in stability theory, system norm estimation, and robust control. A central problem that emerges is the construction of a storage function. It is this problem that brought LMI's to the foreground.

The topic of this talk is distributed dissipative systems. First some basic system theoretic concepts, as variable elimination, controllability and observability, are introduced for systems described by linear constant coefficient PDE's, within the behavioral framework.

Subsequently, dissipative systems described by linear PDE's and supply rates that are quadratic expressions in the system variables and their partial derivatives are defined. The dissipation inequality for such systems involves, in addition to the storage function, also the flux. The construction of the storage and the flux reduces to the factorization of polynomial matrices in many variables. This leads straight to Hilbert's 17-th problem regarding the sum-of-squares representation of nonnegative polynomials in many variables. Throughout the talk, Maxwell's equations will be used as the paradigmatic example.