Optimal Control, Dichotomy, and Closed Range

The paper deals with optimality conditions of variational type for infinite horizon discrete time optimal control problems. We show that for unconstrained problems standard first order conditions extend from a finite to an infinite horizon provided a certain linear operator has closed range. This is the case if the dynamics of the problem has exponential dichotomy, a frequented concept in dynamical systems theory. In addition, closedness of the range allows us to phrase a simple condition for the “normality” of the optimality conditions.