  In this paper we solve the exit problems for an one-sided Markov additive
process (MAP) which is exponentially killed with a bivariate killing intensity
$\omega(\cdot,\cdot)$ dependent on the present level of the process and the
present state of the environment. Moreover, we analyze respective resolvents.
All identities are given in terms of new generalizations of classical scale
matrices for the MAP. We also remark on a number of applications of the
obtained identities to (controlled) insurance risk processes. In particular, we
show that our results can be applied to the so-called Omega model, where
bankruptcy occurs at rate $\omega(\cdot,\cdot)$ when the surplus process
becomes negative. Finally, we consider the Markov modulated Brownian motion
(MMBM) and present the results for the particular choice of piecewise intensity
function $\omega(\cdot,\cdot)$.
