Existence of Optimal Controls for a Bi-Level Optimal Control Problem

Abstract

In this paper we consider a class of bi-level optimal control problems. Both the upper and lower level problems are formulated as ordinary optimal control models of Lagrange type. Our goal is to formulate and prove a general existence theorem for an optimal solution based on classical compactness, convexity and seminormality conditions originating in the work of L. Tonelli for ordinary calculus of variations problems and extended to optimal control problems by L. Cesari, R.T. Rockafellar, L. Berkovitz and others. A distinguishing feature of our result is that we do not require the lower level problem to have a unique optimal solution corresponding to each admissible strategy of the upper level problem.