Path integrals and Lyapunov functionals

A method for generating Lyapunov functionals for time-delay systems by means of path integrals in state space is given. The method is derived by making use of a new description of such systems in terms of convolution equations involving distributions with compact support. The important properties of these equations are discussed and it is shown that a suitable state space can be defined. Path integrals in this state space are defined and conditions for path independence are derived. With the aid of some results dealing with the spectral factorization of entire functions of exponential order, it is shown that these path integrals can be used to define Lyapunov functionals for time-delay systems. The method given represents an extension to infinite-dimensional systems of a technique developed by Brockett for systems described by ordinary differential equations. While the present approach differs fundamentally from that used for finite-dimensional systems, the results given here are similar to, and in the special case of finite-dimensional systems reduce to, the results given by Brockett. Hence the method given can be successfully applied even without a deep understanding of either distributions or distributional convolution equations. This is illustrated by a number of examples which show the application of the results to stability analysis as well as to a class of quadratic minimiization problems.