Interconnections of ${\mathscr {L}}^2$-Behaviors: Lumped Systems

Abstract

J.C. Willems’ fundamental work in electrical circuit theory spawns many questions regarding energy and its transfer across ports. This paper proposes the inverse limit of the Sobolev spaces (on ${\mathbb {R}}$) as the appropriate space of signals in which to address these questions. Some of the first questions are those of interconnections of circuits in this signal space, and of the elimination of latent variables. The answers to these questions in the setting of the Sobolev limit naturally lead to questions on implementability and on the regularity of implementing controllers in the sense of H.L. Trentelman and Willems.