Control problems with kinks

1 October 1970

journal article
Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Automatic Control

Vol. 15 (5) , 570-575
https://doi.org/10.1109/tac.1970.1099557

Abstract

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.

Keywords

This publication has 3 references indexed in Scilit:

Convex Analysis
Published by Walter de Gruyter GmbH ,1970
Non-Linear Programming Via Penalty Functions
Management Science, 1967
Minimax Theorems
Proceedings of the National Academy of Sciences, 1953