In this paper, we are concerned with the basic problem defined in . Formulas for δV(0)and δ
∞V(0),respectively the generalized and asymptotic gradient of the value function at zero, corresponding to an L
2 -additive perturbation of dynamics are given. Under the normality condition, δV(0)turns out to be a compact subset of L2, formed entirely of arcs, and V is locally finite and Lipschitz at 0. Moreover, estimations of the generalized directional derivative and Dini's derivative of V at 0 are derived. Supplementary conditions imply that Dini's derivative of V at 0 exists, and V is actually strictly differentiate at this point.