  An optimal ergodic control problem (EC problem, for short) is investigated
for a linear stochastic differential equation with quadratic cost functional.
Constant nonhomogeneous terms, not all zero, appear in the state equation,
which lead to the asymptotic limit of the state non-zero. Under the
stabilizability condition, for any (admissible) closed-loop strategy, an
invariant measure is proved to exist, which makes the ergodic cost functional
well-defined and the EC problem well-formulated. Sufficient conditions,
including those allowing the weighting matrices of cost functional to be
indefinite, are introduced for finiteness and solvability for the EC problem.
Some comparisons are made between the solvability of EC problem and the
closed-loop solvability of stochastic linear quadratic optimal control problem
in the infinite horizon. Regularized EC problem is introduced to be used to
obtain the optimal value of the EC problem.
