Computation of closed loop eigenvalues associated with the optimal regulator problem for functional differential equations

A solution of the linear quadratic control problem involving functional differential equations gives a linear feedback control law which modifies the original system dynamics. Under certain assumptions, the eigenvalues of the modified linear system constitute a stable part of a spectrum of a hamiltonian operator associated with the optimization problem. These eigenvalues can be computed without solving the infinite dimensional Riccati equation. In this paper we present a method based on an earlier algorithm (constructed by A. Manitius, G. Payre and R. Roy) which solves directly the characteristic equation of the closed loop system, and compare it with a direct computation of eigenvalues of a symplectic hamiltonian matrix arising from a finite dimensional approximation of a functional differential equation.