Existence of a value in semidynamical games

The games (generalizing differential games) in which the dynamics of players is described by $k$-semidynamical systems are called semidynamical games. For such games two theorems on the existence of a value in the class of piecewise program strategies are proved. Examples are given to show that the conditions of these theorems impose very weak restrictions on the set of admissible controls of the players and, in the games with a fixed duration, on the set of trajectories of semidynamical systems.