  We introduce the Lyapunov approach to optimal control problems of average
risk-sensitive Markov control processes with general risk maps. Motivated by
applications in particular to behavioral economics, we consider possibly
non-convex risk maps, modeling behavior with mixed risk preference. We
introduce classical objective functions to the risk-sensitive setting and we
are in particular interested in optimizing the average risk in the
infinite-time horizon for Markov Control Processes on general, possibly
non-compact, state spaces allowing also unbounded cost. Existence and
uniqueness of an optimal control is obtained with a fixed point theorem applied
to the nonlinear map modeling the risk-sensitive expected total cost. The
necessary contraction is obtained in a suitable chosen seminorm under a new set
of conditions: 1) Lyapunov-type conditions on both risk maps and cost functions
that control the growth of iterations, and 2) Doeblin-type conditions, known
for Markov chains, generalized to nonlinear mappings. In the particular case of
the entropic risk map, the above conditions can be replaced by the existence of
a Lyapunov function, a local Doeblin-type condition for the underlying Markov
chain, and a growth condition on the cost functions.
