ON QUADRATIC DIFFERENTIAL FORMS

This paper develops a theory around the notion of quadratic differenti al forms in the context of linear differential systems. In many applic ations, we need to not only understand the behavior of the system vari ables but also the behavior of certain functionals of these variables. The obvious cases where such functionals are important are in Lyapuno v theory and in LQ and H-infinity optimal control. With some exception s, these theories have almost invariably concentrated on first order m odels and state representations. In this paper, we develop a theory fo r linear time-invariant differential systems and quadratic functionals . We argue that in the context of systems described by one-variable po lynomial matrices, the appropriate tool to express quadratic functiona ls of the system variables are two-variable polynomial matrices. The m ain achievement of this paper is a description of the interaction of o ne- and two-variable polynomial matrices for the analysis of functiona ls and for the application of higher order Lyapunov functionals.