Linear-quadratic-Gaussian differential games with different information patterns

A discrete-time linear-quadratic-Gaussian (LQG) differential game is considered where the players have access to separate measurement histories. The particular problem that is solved is where one adversary has access to only noisy partial information of the state while the other makes a perfect measurement of the state vector. The system dynamics are assumed linear with additive process noise. The solutions show a significant departure from previously published results. First, process noise is included in the dynamical system and a quadratic weighting in the state is included in the cost criterion. Secondly, the optimal strategies of both players are shown to be finite dimensional, not infinite dimensional as was originally thought. There is, therefore, no reason for the player with perfect measurement to have additional information to reconstruct the other players measurement. Thirdly, it is assumed that the perfect-measurement adversary´s control matrix is in the range space of the other adversary´s measurement matrix. Then, by a limit of the linear-exponential-Gaussian game solution to the LQG game solution, it is seen that the partial information player avoids reproducing an estimated version of his adversary´s strategy.