Necessary Conditions without Differentiability Assumptions in Unilateral Control Problems*

We derive two theorems combining existence with necessary conditions for the relaxed unilateral problem of the optimal control of ordinary differential equations in which the functions that define the problem are Lipschitz-continuous in the state variables. These theorems generalize the results presented in a previous paper [8] by the addition of unilateral constraints on the state and control functions. As in that paper, the new necessary conditions have a canonical form obtained by replacing, in the “customary” conditions, the partial derivatives with respect to the state variables by finite difference quotients at neighboring arguments, and then applying limiting processes and convexification. More general necessary conditions are also obtained in terms of the representations of the Lipschitz-continuous functions as compositions,