Linear quadratic differential games: Saddle point and Riccati differential equation

Zhang [ SIAM J. Control Optim., 43 ( 2005), pp. 2157-2165] recently established the equivalence between the finiteness of the open loop value of a two-player zero-sum linear quadratic ( LQ) game and the finiteness of its open loop lower and upper values. In this paper we complete and sharpen the results of Zhang for the finiteness of the lower value of the game by providing a set of necessary and sufficient conditions that emphasizes the feasibility condition: ( 0, 0) is a solution of the open loop lower value of the game for the zero initial state. Then we show that, under the assumption of an open loop saddle point in the time horizon [ 0, T] for all initial states, there is an open loop saddle point in the time horizon [ s, T] for all initial times s, 0 <= s < T, and all initial states at time s. From this we get an optimality principle and adapt the invariant embedding approach to construct the decoupling symmetrical matrix function P( s) and show that it is an H-1( 0, T) solution of the matrix Riccati differential equation. Thence an open loop saddle point in [ 0, T] yields closed loop optimal strategies for both players. Furthermore, a necessary and sufficient set of conditions for the existence of an open loop saddle point in [ 0, T] for all initial states is the convexity-concavity of the utility function and the existence of an H-1( 0, T) symmetrical solution to the matrix Riccati differential equation. As an illustration of the cases where the open loop lower/upper value of the game is -infinity/+infinity, we work out two informative examples of solutions to the Riccati differential equation with and without blow-up time.