Application of abstract variational theory to hereditary systems--A survey

In this survey paper, results for optimal control problems governed by hereditary systems are presented and discussed. Although fundamental results (controllability, existence and uniqueness of optimal controls, feedback controls) obtained without employing abstract variational approaches are briefly reviewed, emphasis is placed on those results established through the use of general abstract variational theories, and, in particular, on those growing out of the theories developed by the late Lucien W. Neustadt. Motivating examples of hereditary systems encountered in applications (technological, biological, and biomedical) are given. Abstract variational theories involving the Lagrange Multiplier rule and the Kuhn-Tucker conditions in a Banach space, the approach of Dubovitskii-Milyutin, the quasiconvexity ideas of Gamkrelidze, and the abstract maximum principles developed in the work of Halkin and Neustadt are summarized briefly. A more detailed discussion of applications of the quasiconvexity ideas and the Neustadt theory to systems governed by differential-difference, functional-differential, and Volterra integral equations is presented for control problems both with and without state constraints. Recent results for control of hereditary systems with terminal function space boundary conditions are also reviewed.