The sum-of-squares problem and dissipative systems

In this presentation, we discuss the theory of dissipativeness of systems described by linear constant coefficient PDE's with respect to supply rates that are quadratic differential forms in the variables and their derivatives. The main issue considered is the equivalence of global and local dissipativeness. This leads to the construction of the storage function, the flux, and the dissipation rate. We show that mathematically this leads to Hilbert's 17-th problem on the factorization of a polynomial in n variables as a sum of squares.