  Passivity property gives a sense of energy balance. The classical definitions
and theorems of passivity in dynamical systems require time invariance and
locally Lipschitz functions. However, these conditions are not met in many
systems. A characteristic example is nonautonomous and discontinuous systems
due to presence of Coulomb friction. This paper presents an extended result for
the negative feedback connection of two passive nonautonomous systems with
set-valued right-hand side based on an invariance-like principle. Such
extension is the base of a structural passivity-based control synthesis for
underactuated mechanical systems with Coulomb friction. The first step consists
in designing the control able to restore the passivity in the considered
friction law, achieving stabilization of the system trajectories to a domain
with zero velocities. Then, an integral action is included to improve the
latter result and perform a tracking over a constant reference (regulation). At
last, the control is designed considering dynamics in the actuation. These
control objectives are obtained using fewer control inputs than degrees of
freedom, as a result of the underactuated nature of the plant. The presented
control strategy is implemented in an earthquake prevention scenario, where a
mature seismogenic fault represents the considered frictional underactuated
mechanical system. Simulations are performed to show how the seismic energy can
be slowly dissipated by tracking a slow reference, thanks to fluid injection
far from the fault, accounting also for the slow dynamics of the fluid's
diffusion.
