Linear-Exponential-Quadratic Gaussian Control

In this technical note an optimal control problem for a linear stochastic system with Brownian motion and a cost that is an exponential of a quadratic functional of the state and the control is solved by obtaining explicitly an optimal control and the optimal cost. While this solution has been previously obtained, the approach given here is direct and elementary and does not use the well known solution methods of the Hamilton-Jacobi-Bellman equation or the stochastic maximum principle. The approach given here presents a basic insight in the solution by providing a simple explanation for the additional term in the Riccati equation for the optimal control as compared to the Riccati equation for the linear-quadratic Gaussian control problem.