A rate of approach to the steady state of solutions of second-order hyperbolic equations

In this paper projection methods based on expansions of solutions of retarded function differential equations in terms of generalized eigenfunctions are considered. It is first shown that the projection series developed earlier by Hale and Shimanov and those considered by Bellman and Cooke are actually the same. Using extensions of the residue-type arguments of Bellman and Cooke, convergence results are then established for a class of perturbed systems. These results are applied to obtain approximations to optimal controls for certain infinite dimensional variational problems. Numerical results are presented for several examples.