An Abstract Framework for Approximate Solutions to Optimal Control Problems Governed by Hereditary Systems

General projection methods for nonhomogeneous linear functional differential equations (FDE) dx/dt = L(x sub t) + f(t) are considered. It is shown that by an appropriate choice of spaces these can be treated as an equivalent abstract equation in a Banach space where the solution of the homogeneous equation generates a strongly continuous semigroup. Well-known approximation results for such semigroups (e.g., Trotter's theorem) are used to develop a general framework for obtaining approximate solutions of the original FDE. Convergence results are easily established. Applications to optimal control of linear FDE are discussed. In particular, the original infinite-dimensional problem is replaced by a 'projected' finite-dimensional problem which is more easily solved. Convergence of solutions of these finite-dimensional problems to solutions of the original problem is obtained.
Sponsored in part by National Science Foundation, Washington, D.C. Grant NSF-GP-28931.