Periodic existence theorems in optimal control

Abstract

This paper considers existence theorems for the optimal control problem min (x,u) lo(x(0),x(T))+ ʃT0 fo(t,x(t),x(t),u(t)) subject to: x(t) = f(t ,x(t),u(t)) (x(0), x(T)) ∈ B (x(t),u(t)) ∈ F x ∈ An[0, T] u ∈ Lm[0, T] Using the results of R.T. Rockafellar this problem is reformulated into a problem in which the control and the constraints are absorbed into the objective function. Rockafellar has established a general existence theorem for such control problems. This existence theorem requires finding bounds on a Hamiltonian function. In his work Rockafellar specialized his results to certain initial value problems. The central new theorems in this paper specialize Rockafellar's general theorem to periodic problems in optimal control. These theorems are then coupled with results of R.E. Gaines and J. Peterson which give necessary conditions for a finite minimum.