One Aspect On Bellman's Equation To Optimal Control Theory ∗

In this present work, we develop the idea of the dynamic programming ap-proach. The main observation is that the Bellman function ω(x, t) which is the function that provides, for any given state x at any given time t, the smallest possible cost among all possible trajectories starting at this event is in general not differentiable, and consequently we cannot use the Hamilton-Jacobi-Bellman (HJB)equation. By the classical Hamilton-Jacobi-Bellman theory if the value function ω is continuously differentiable, then it is the unique solution of the (HJB) equation. It is well known that the value function ω is in general discon-tinuous, even if all the data of the problem is continuously differentiable. Using the (HJB) equation in some nonclassical sense (e.g. using generalized gradients, in the framework of viscosity solutions, proximal solutions, etc.) has become a very active research area. Here, we give techniques based on the differential set ∂ω(x, l) of the function ω at the element x along the direction l for the analysis of such problems. 1 Description of the problem Let X a Banach space, B a closed unit ball in X, W a nonempty subset of X. Let l ∈ X, x ∈ W. Assume that l, x are such that x + δl ∈ W for δ small enough. Let ρ:W − → R. We define ∂ρ(x, l) as follows: ∂ρ(x, l) =