  In most real cases transition probabilities between operational modes of
Markov jump linear systems cannot be computed exactly and are time-varying. We
take into account this aspect by considering Markov jump linear systems where
the underlying Markov chain is polytopic and time-inhomogeneous, i.e. its
transition probability matrix is varying over time, with variations that are
arbitrary within a polytopic set of stochastic matrices. We address and solve
for this class of systems the infinite-horizon optimal control problem. In
particular, we show that the optimal controller can be obtained from a set of
coupled algebraic Riccati equations, and that for mean square stabilizable
systems the optimal finite-horizon cost corresponding to the solution to a
parsimonious set of coupled difference Riccati equations converges
exponentially fast to the optimal infinite-horizon cost related to the set of
coupled algebraic Riccati equations. All the presented concepts are illustrated
on a numerical example showing the efficiency of the provided solution.
