The Application of Crossed Products to the Stability and Design of Time- Varying Systems

Abstract

Abstract : The research conducted for the past two years into linear time- varying systems is outlined. Corssed product algebras make precise the intuitive idea of a class of systems which can be synthesized from the usual delay elements and a fixed class of time-varying gains. It has been shown that crossed products appear to be a most appropriate setting for input-output analysis of linear time-varying dynamical systems; they also always admit a bounded decomposition of a Hermitian operator into the sum of a causal operator and its adjoint. Finally it has been shown that in the context of crossed products it is possible (with a restriction on the class of time-varying gains) to formulate a generalized transform theory which closely parallels that for time-invariant systems, and yields previously known results for periodic systems.