REVIEW OF DIRECT SUFFICIENT CONDITIONS IN OPTIMAL-CONTROL THEORY

The development of conditions sufficient to ensure that a proposed solution to a continuous-time optimal control problem is indeed a solution has proceeded along two general lines. One of these. with origins in field theory, is identified with the works of Weierst.rass, Hamilton, Jacobi and Bellman; the other line, which we call the direct theory, is related to the saddle-point theory of mathematical programming, to Pontryngin's theory of optimal processes, and to some extent to the vurlut.ional calculus. The direct sufficient conditions are the subject of this paper. It is found that four important direct sufficient conditions arc of varying scope and that one of these conditions subsumes the others. The differences among the conditions are illustrated by both theorem and example. and speculations arc offered on the possibilities for the development of additional direct sufficient conditions. The most powerful results found are mustered in the paper's final theorem. © 1978 Taylor & Francis Group, LLC.