Necessary and sufficient conditions for optimally for singular control problems; a transformation approach

Abstract

Necessary and sufficient conditions for optimality for singular control problems are presented for the case where the extremal path is totally singular. These conditions are obtained by transforming the perturbed states of the accessory minimum problem in the calculus of variations to a new set of state variables in which the dimension of the state space can be reduced. The accessory problem in the reduced state space is nonsingular, and thus previously established necessary and sufficient conditions are applicable. It is shown that Jacobson's sufficiency conditions for singular control problems applied to the total transformed state space are equivalent to those of the nonsingular accessory problem thereby establishing necessity as well as sufficiency for Jacobson's conditions. In fact, since the transformation is nonsingular, necessity as well as sufficiency of Jacobson's conditions is established in the original space. In a companion paper a limit argument is used to establish necessity of Jacobson's sufficient conditions without the need to transform to a reduced state space.