a fairly explicit procedure which relies on the proof of superoptimality inequality of dynamic programming for viscosity supersolutions of the corresponding Hamilton-Jacobi-Bellman equation. It is a delicate generalization of such a method for the finite dimensional case from [@sw]. Similar method has been used in [@CLSS] to construct stabilizing feedbacks for nonlinear systems and later in [@IK] for state constraint problems. The idea here is to approximate the value function by its appropriate inf-convolution which is more regular and satisfies a slightly perturbed HJB inequality pointwise. One can then use this inequality to construct $\epsilon$-optimal piecewise constant controls. This procedure in fact gives the superoptimality inequality of dynamic programming and the suboptimality inequality can be proved similarly. There are other possible approaches to construction of $\epsilon$-optimal controls. For instance under compactness assumption on the operator $B$ (see Section 4) one can approximate the value function by solutions of finite dimensional HJB equations with the operator $A$ replaced by some finite dimensional operators $A_n$ (see [@CL4]) and then use results of [@sw] directly to construct near optimal controls. Other approximation procedures are also possible. The method we present in this paper seems to have some advantages: it uses only one layer 