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                <term xml:lang="en">linear-quadratic stochastic control</term>
                <term xml:lang="en">long-term average cost</term>
                <term xml:lang="en">Riccati equation</term>
                <term xml:lang="en">ergodicity</term>
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              <p>We consider a stochastic control problem with time-inhomogeneous linear dynamics and a long-term average quadratic cost functional. We provide sufficient conditions for the problem to be well-posed. We describe an explicit optimal control in terms of a bounded and non-negative solution of a Riccati equation on $[0, \infty)$, without an initial and terminal condition. We show that, in contrast to the time-homogeneous case, in the inhomogeneous case the optimally controlled state dynamics are not necessarily ergodic.</p>
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