  We consider continuous-time, finite-horizon, optimal quadratic control of
semi-Markov jump linear systems (S-MJLS), and develop principled approximations
through Markov-like representations for the holding-time distributions. We
adopt a phase-type approximation for holding times, which is known to be
consistent, and translates a S-MJLS into a specific MJLS with partially
observable modes (MJLSPOM), where the modes in a cluster have the same dynamic,
the same cost weighting matrices and the same control policy. For a general
MJLSPOM, we give necessary and sufficient conditions for optimal (switched)
linear controllers. When specialized to our particular MJLSPOM, we additionally
establish the existence of optimal linear controller, as well as its optimality
within the class of general controllers satisfying standard smoothness
conditions. The known equivalence between phase-type distributions and positive
linear systems allows to leverage existing modeling tools, but possibly with
large computational costs. Motivated by this, we propose matrix exponential
approximation of holding times, resulting in pseudo-MJLSPOM representation,
i.e., where the transition rates could be negative. Such a representation is of
relatively low order, and maintains the same optimality conditions as for the
MJLSPOM representation, but could violate non-negativity of holding-time
density functions. A two-step procedure consisting of a local pulling-up
modification and a filtering technique is constructed to enforce
non-negativity.
