Optimal control of nonlinear hereditary systems

A function space Λ is introduced for the study of nonlinear hereditary differential equations. The properties of Λ include: it is a Banach space under the supremum norm, the continuous functions constitute a closed proper subspace, and the unit ball is sequentially compact in the weak-¹ topology. Existence, uniqueness, and continuous dependence results are obtained for solutions of a broad class of initial value problems. An optimization problem is formulated for systems which are affine in the control, and solutions are approximated by means of a sequence of problems which are finite-dimensional in the control.