Control problems with kinks

An important class of optimal control problems, arising frequeutly in an economic framework, is characterized as having a cost functional that is continuous but has discontinuous partial derivatives with respect to the state variables. Such problems are said to have kinks. Along a kink the classical adjoint equation breaks down, and it is impossible to define a gradient. In this paper it is shown that the gradient can be replaced by a more general definition of the direction of steepest descent but that the adjoint equation must in general be replaced by an adjoint optimal control problem. This yields a complete set of necessary conditions for problems of this type. The results derived are then combined with the theory of penalty functions to convert a problem having state constraints to one without such constraints.