On the Riccati Partial Differential Equation for Nonlinear Bolza and Lagrange Problems

In this paper, we study the solution of certain optimal control problems by the use of an off-line Riccati Partial Differential Equation which is shown to be equivalent to the Hamilton-Jacobi-Bellman (HJB) equation, when the value function is sufficiently smooth. We develop a geometric existence theory for classical, weak and generalized solutions of the Riccati Partial Differential Equation. For finite and infinite time hori-zon problems, we also investigate the relationship of such solutions to optimal control laws. Indeed, as corollaries of certain of our existence results, we obtain smoothness results for the value function of the corresponding optimal control problem. 1