  Bellman equations of ergodic type related to risk-sensitive control are
considered. We treat the case that the nonlinear term is positive quadratic
form on first-order partial derivatives of solution, which includes linear
exponential quadratic Gaussian control problem. In this paper we prove that the
equation in general has multiple solutions. We shall specify the set of all the
classical solutions and classify the solutions by a global behavior of the
diffusion process associated with the given solution. The solution associated
with ergodic diffusion process plays particular role. We shall also prove the
uniqueness of such solution. Furthermore, the solution which gives us
ergodicity is stable under perturbation of coefficients. Finally, we have a
representation result for the solution corresponding to the ergodic diffusion.
