  We consider a scenario where a system experiences a disruption, and the
states (representing health values) of its components continue to reduce over
time, unless they are acted upon by a controller. Given this dynamical setting,
we consider the problem of finding an optimal control (or switching) sequence
to maximize the sum of the weights of the components whose states are brought
back to the maximum value. We first provide several characteristics of the
optimal policy for the general (fully heterogeneous) version of this problem.
We then show that under certain conditions on the rates of repair and
deterioration, we can explicitly characterize the optimal control policy as a
function of the states. When the deterioration rate (when not being repaired)
is larger than or equal to the repair rate, and the deterioration and repair
rates as well as the weights are homogeneous across all the components, the
optimal control policy is to target the component that has the largest state
value at each time step. On the other hand, if the repair rates are
sufficiently larger than the deterioration rates, the optimal control policy is
to target the component whose state minus the deterioration rate is least in a
particular subset of components at each time step.
