Factorization and the synthesis of optimal feedback kernels for differential-delay systems

In this paper a combination of ideas from the theories of operator Riccati equations and Volterra factorizations leads to the derivation of a new and relatively simple set of hyperbolic equations which characterize the optimal feedback kernel for the finite-time regulator problem for autonomous differential-delay systems. Analysis of these equations elucidates the underlying structure of the feedback kernel and leads to the development of fast and accurate numerical methods for its computation. Unlike traditional formulations based on the operator Riccati equation, the gain is characterized by means of classical solutions of the derived set of equations. This leads to the development of approximation schemes which are analagous to what has been accomplished for systems of ordinary differential equations with given initial conditions.