A type of linear game with mixed constraints on the controls (Q1110465)

A game is considered in which the first person's control is subject to a geometric and an integral constraint. The second person can choose a control, subject to the geometric constraint. The game ends at a given instant. The sets of control values and the terminal set are of the same type. The conditions are found for the game to end from a given initial position. The player's controls are constructed, and an example is given. Differential games with mixed constraints were considered by \textit{G. K. Pozharitskij} [J. Appl. Math. Mech. 46, 311-320 (1983; Zbl 0518.90106)] and \textit{Yu. S. Ledyaev} [Tr. Mat. Inst. Steklova 167, 207-215 (1985; Zbl 0597.90100)]. The present types of game may not satisfy the regularity condition of the second paper mentioned above.