Necessary Conditions for Optimality for Paths Lying on a Corner

A class of optimization problems is investigated in which some of the functions, continuous in all their arguments, have continuous right- and left-hand derivatives but are not equal at a point called the corner. For this nonclassical problem, a set of first order necessary conditions for stationarity is determined for an optimal path which may have arcs lying on a corner for a nonzero length of time. Enough conditions are provided to construct an extremal path. This, in part, is achieved by noting that the corner defines a manifold in which the derivatives of all the functions are uniquely defined. Three examples, two of which represent possible aggregate production and employment planning models, illustrate the theory.