Linear quadratic differential game problems with constrained open-loop controls

Recently it is established that the finiteness of the open-loop value of a two-player zero-sum linear quadratic differential game is equivalent to the finiteness of its open-loop lower and upper values (P. Zhang, SIAM J. Control Optim., 43(2005), pp. 2157-2165). In this paper we discuss the case that the control sets are restricted in some elliptic regions, under which the linear operator method fails. By studying a two-point boundary value problem and the necessary conditions for an extremum, we prove that the value of the game exists if and only if both the upper and lower values of the unconstrained problem exist, which implies the results of Zhang. Further, the expression of the value function is obtained.