On the problem of the fastest response

The paper is concerned with continuity, the Lipschitz condition, and differentiability of the Bellman function in the classical problem of fastest response [1-3] with a single-point and convex objective set. The paper extends the findings of Ref. [4]. Conditions are obtained under which the law from the motion of the controlled system, the region of controlled parameters, and the objective set the presence or absence of these properties of the Bellman function can be linearly ascertained some considerations on nonlinear problems are given. The resultant conditions lead to a simple way of smoothing the Bellman function in linear problems with a single-point objective set which removes [2] the mathematical difficulties of using the Bellman dynamic programming method in its classical formulation [1]. Examples are given. The bulk of the article is given chiefly to continuity of the Bellman function.