Existence results without convexity conditions for general problems of optimal control with singular components

This note presents a new, quick approach to existence results without convexity conditions for optimal control problems with singular components in the sense of E. J. McShane (SIAM J. Control5 (1967), 438–485). Starting from the resolvent kernel representation of the solutions of a linear integral equation, a version of Fatou's lemma in several dimensions is shown to lead directly to a compactness result for the attainable set and an existence result for a Mayer problem. These results subsume those of L. W. Neustadt (J. Math. Anal. Appl.7 (1963), 110–117), C. Olech (J. Differential Equations2 (1966), 74–101), M. Q. Jacobs (“Mathematical Theory of Control,” pp. 46–53, Academic Press, 1967), L. Cesari (SIAM J. Control12 (1974), 319–331) and T. S. Angell (J. Optim. Theory Appl.19 (1976), 63–79).