Shifting the closed-loop spectrum in the optimal linear quadratic regulator problem for hereditary systems

In the optimal linear quadratic regulator problem for finite-dimensional systems, the method known as an α-shift can be used to produce a closed-loop system whose spectrum lies to the left of some specified vertical line; that is, a closed-loop system with a prescribed degree of stability. This note treats the extension of the α-shift to hereditary systems. As in finite dimensions, the shift can be accomplished by adding α times the identity to the open-loop semigroup generator and then solving an optimal regulator problem. However, this approach does not work with a new approximation scheme for hereditary control problems recently developed by Kappel and Salamon. Since this scheme is among the best to date for the numerical solution of the linear regulator problem for hereditary systems, an alternative method for shifting the closed-loop spectrum is needed. An α-shift technique that can be used with the Kappel-Salamon approximation scheme is developed. A numerical example which demonstrates the feasibility of the method is included.