The notion of a dissipative dynamical system was introduced in the early 1970’s. It 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 main aim of this talk is to discuss distributed dissipative systems. We will first introduce some basic system theoretic concepts for systems described by linear constant coefficient PDE’s, within the behavioral framework. Issues as sub-module characterizations, controllability, and observability will be introduced.
We will then turn our attention to dissipative systems described by linear PDE’s and supply rates that are quadratic expressions in the system variables and their partial derivatives. The dissipation inequality for dissipative distributed systems now involves the storage and the flux. Their construction reduces to factorization of polynomial matrices in many variables. This leads straight to Hilbert’s 17-th problem and the sum-of-squares problem. Throughout the talk, we will take Maxwell’s equations as our paradigmatic example.
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