A formula for the optimal cost in the general discrete-time LEQG problem

The study of linear control problems with exponential-ofquadratic cost function was initiated in Jacobson (1973) where the author studies discrete-time and continuoustime state feedback problems. The discrete-time linear output feedback problems with exponential-of-quadratic cost function is studied in Whittle (1981). Bensousssan and Schuppen (1985) studied the continuous-time output feedback problem in this context, and, in Pan and Basar (1996), these results were given simpler proofs for the general case with cross terms in cost function and noise covariance matrices. Furthermore, an elegant expression for the optimal cost is derived in Pan and Basar (1996) which has been subsequently used to obtain explicit guaranteed cost bounds for continuous-time uncertain systems and to solve minimax LQG control problems (e.g., see Petersen et al. (2000)). This expression was also instrumental in the guaranteed cost approach to nonlinear control and state estimation presented in the papers Petersen (2006b,a, 2007); Ouyang and Petersen (2007).