Remarks on some existence theorems for optimal control

First, a remark is made that a growth condition contained in previous papers by Cesari concerning existence theorems for optimal controls can be replaced by a slightly more general condition. In this more general condition, a constantM 0 is replaced by any functionM (t)0 which is assumed to beL-integrable in every finite interval.Then, the remark is made that the same condition, which is usually required to be satisfied by the functionsf ₀(t, x, u),f(t, x, u) characterizing the control, can be required to be satisfied only by the admissible pairsx(t),u(t) of the class in which the optimum is being sought. This generalization requires a subtle argument. The new condition parallels now the usual conditions of the type |x| ^p dtM, which are required to be satisfied by the admissible pairs of the class .The work of L. Cesari was partially supported by the Office of Scientific Research, United States Air Force, Grant No. 942-65. The work of T. Nishiura was partially supported by the National Science Foundation, Grant No. GP-7676.