Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games

Two stochastic optimal control problems are solved whose performance criteria are the expected values of exponential functions of quadratic forms. The optimal controller is linear in both cases but depends upon the covariance matrix of the additive process noise so that the certainty equivalence principle does not hold. The controllers are shown to be equivalent to those obtained by solving a cooperative and a noncooperative quadratic (differential) game, and this leads to some interesting interpretations and observations. Finally, some stability properties of the asymptotic controllers are discussed.