Construction of the players best guaranteeing strategy in one antagonistic game with non-differentiable value

In this paper a differential directing-evading game on horizontal plane in the class of positional strategies is discussed. Its value is demonstrated to be a continuously differentiable function not for every position. In the paper, in order to implement an optimal strategy of the first player his stable bridge is constructed so that its section coincides with the target set at final instant of time. Optimal control is carried out by the player in the form of extremal targeting at the constructed bridge.