Suboptimal Poisson checking processes with a bounded loss function

We discuss the problem of minimizing the expected cost associated with a Poisson checking process with a nondecreasing intensity. The checks should detect the possible first failure of a system. In case of failure, the loss associated with it increases linearly with the detection delay, as the delay is less than a given threshold, whereas the loss is constant if the delay is greater than the threshold. We consider an optimal control problem which is an approximation of the original one. We prove that an optimal solution exists, discuss the necessary conditions obtained from the Pontryagin’s Maximum Principle and provide an algorithm to determine the optimal solution numerically.