On the problem of synthesis of piece-wise mixed-type strategies

A method of building piece-wise mixed-type strategies is studied in differential games by means of stochastic programming synthesis for non-linear systems. It is proved that the introduction of the auxiliary phase coordinate makes possible to build rationally such a function, which approximates the value of the game and is differentiable on all phase coordinates. In this case the optimal strategy is defined through the condition that the extremal replacement must be opposite to the gradient of the approximation function.