Solvable risk-sensitive control problems with output feedback

In this paper a partially observed nonlinear stochastic control with exponential running cost is considered. Explicit solutions for evaluating expectations of exponential functions are found when the control is assumed to be constant. This enables the above problems to be expressed in terms of observable quantities, while the separation principle applies. First, we consider the case when the dynamics are linear but the observations are quadratic functions of the signal. Second, we consider the case when the dynamics are nonlinear, but the observations are linear functions of the signal. Although in the present paper only control systems with constant controls are considered, the results remain applicable to more general control systems. The results presented here, are helpful in evaluating expectations of exponential functionals, similar to Feynman Kac functionals, when the dynamics are autonomous, nonlinear, and partially observable.