  We discuss the problem of optimal impulse control representing the preventive
maintenance of a simple reparable system. The system model is governed by
coupled transport and integro-differential equations in a nonreflexive Banach
space. The objective of this paper is to construct nonnegative impulse control
inputs at given system running times that minimize the probability of the
system in failure mode. To guarantee the nonnegativity of the controlled
system, we consider the control inputs to depend on the system state. This
essentially leads to a bilinear control problem. We first present a rigorous
proof of existence of an optimal controller and then apply the variational
inequality to derive the first-order necessary conditions of optimality.
