Well-posed systems - The LTI case and beyond

This survey is an introduction to well-posed linear time-invartiant (LTI) systems for non-specialists. We recall the more general concept of a system node, classical and generalized solutions of system equations, criteria for well-posedness, the subclass of regular linear systems, some of the available linear feedback theory. Motivated by physical examples, we recall the concepts of impedance passive and scattering passive systems, conservative systems and systems with a special structure that belong to these classes. We illustrate this theory by examples of systems governed by heat and wave equations. We develop local and global well-posedness results for LTI systems with nonlinear (in particular, bilinear) feedback, by extracting the abstract idea behind various proofs in the literature. We apply these abstract results to derive well-posedness results for the Burgers and Navier-Stokes equations.