  We present a numerical method to compute an optimal maintenance date for the
test case of the heated hold-up tank. The system consists of a tank containing
a fluid whose level is controlled by three components: two inlet pumps and one
outlet valve. A thermal power source heats up the fluid. The failure rates of
the components depends on the temperature, the position of the three components
monitors the liquid level in the tank and the liquid level determines the
temperature. Therefore, this system can be modeled by a hybrid process where
the discrete (components) and continuous (level, temperature) parts interact in
a closed loop. We model the system by a piecewise deterministic Markov process,
propose and implement a numerical method to compute the optimal maintenance
date to repair the components before the total failure of the system.
