  This paper investigates a stochastic linear-quadratic (SLQ, for short)
control problem regulated by a time-invariant Markov chain in infinite horizon.
Under the $L^2$-stability framework, we first study a class of linear backward
stochastic differential equations (BSDE, for short) in infinite horizon. And
then, we discuss the open-loop and closed-loop solvabilities of the SLQ
problem. The open-loop solvability is characterized by the solvability of a
system of coupled forward-backward stochastic differential equations (FBSDE,
for short) in infinite horizon and the convexity of the cost functional, and
the closed-loop solvability is shown that to be equivalent to the open-loop
solvability, which in turn is equivalent to the existence of a static
stabilizing solution to the associated constrained coupled algebra Riccati
equations (CAREs, for short). Using the Hilbert space method, we provide a
sufficient verifiable condition to ensure the SLQ problem is solvable. Finally,
we also solve a class of discounted SLQ problems and give two concrete examples
to illustrate the results developed in the earlier sections.
