Path integrals and Bézoutians for pseudorational transfer functions

There is an effective way of constructing a Lyapunov function without recourse to a state space construction. This is based upon an integral of special type called a path integral, and this approach is particularly suited for behavior theory. The theory successfully exhibits a deep connection between Lyapunov theory and Bezoutians, but it remained mostly in the finite-dimensional context. This paper extends the theory to a class of systems described by a wider class of transfer functions called pseudorational, which contains an interesting class of distributed parameter systems, e.g., delay systems. The paper extends the notion of path integrals using an convolution algebra of distributions, and then relates this theory to an infinite-dimensional version of Bezoutians, which in turn gives rise to a new interesting class of Lyapunov functions.