Lossless and Dissipative Distributed Systems

This paper deals with linear shift-invariant distributed systems. By this we mean systems described by constant coefficient linear partial differential equations. We define dissipativity with respect to a quadratic differential form, i.e., a quadratic functional in the system variables and their partial derivatives. The main result states the equivalence of dissipativity and the existence of a storage function or a dissipation rate. The proof of this result involves the construction of the dissipation rate. We show that this problem can be reduced to Hilbert's 17th problem on the representation of a nonnegative rational function as a sum of squares of rational functions. Key words. quadratic differential forms, linear multidimensional systems, behavioral theory, polynomial matrices, lossless systems, positivity, dissipativeness, storage functions AMS subject classifications. 93A30, 93C20, 13P05, 35G05, 37L99, 35L65 PII. S0363012900368028

• Type: Journal
• Year: 2002
• Where: SIAMCO
• Authors: Harish K. Pillai, Jan C. Willems
• Keywords: Dissipation Rate, Rational Functions, SIAMCO 2002, Storage Functions