ON THE EXISTENCE OF A SOLUTION TO THE DUAL PARTIAL DIFFERENTIAL EQUATION OF DYNAMIC PROGRAMMING FOR GENERAL PROBLEMS OF BOLZA AND LAGRANGE

A main theorem which deals with the existence of a minimum solution to the dual partial differential equation of dynamic programming for optimal control problems of Bolza and Lagrange is proved. An example illustrates the value of this theorem is given. Properties of the value function and dual value function for problems of Bolza and Lagrange are described. Moreover, for these problems the existence of a maximum solution to the partial differential equation of dynamic programming, which satisfies the Lipschitz condition and which is also the value function is presented.