Necessary and sufficient conditions for optimality for singular control problems: A limit approach

Abstract

Necessary and sufficient conditions for optimality for singular control problems are presented for the case where the extremal path is totally singular. The singular second variation is converted into a nonsingular one by addition of a quadratic functional of the control; a parameter 1 ϵ multiplies this added functional. By allowing ϵ to approach infinity the optimality conditions are deduced for the singular problem from the limiting optimality conditions of the synthesized nonsingular second variation. The resulting conditions are Jacobson's sufficient conditions in slightly modified form. In a companion paper necessity of Jacobson's conditions for a class of singular problems is demonstrated by exploiting the Kelley transformation technique which converts the singular second variation into a nonsingular one in a reduced dimensional state space.