The sufficient conditions for continuous ɛ-optimal feedbacks in control problems with a terminal cost

The existence of continuous positional strategies of ɛ-optimal feedback is proved for linear optimal control problems with a convex terminal cost. These continuous feedbacks are determined from Bellman's equation in ɛ-perturbed control problems with an integral-terminal cost and a smooth value function. An example is given in which an ɛ-optimal continuous feedback does not exist. It is shown that the point limit of the ɛ-optimal feedbacks when ɛ→0 determines the optimal feedback, that is, a positional strategy and, possibly, a discontinuous strategy.