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 feeback which modifies the original system dynamics. Under certain assumptions, the eigenvalues of the modified system constitute a stable part of the spectrum of a certain Hamiltonian operator. These eigenvalues can be computed without solving the infinite-dimensional Riccati equation. In this note we present a method to compute the eigenvalues directly from a characteristic equation of the optimal closed-loop system. Numerical results are presented for three examples and compared to those obtained by a finite-dimensional approximation of a functional differential equation.