Slow and heavy viable trajectories of controlled problems. Smooth viability domains

Abstract

We define slow and heavy viable trajectories of differential inclusions and controlled problems. Slow trajectories minimize at each time the norm of the velocity of the state (or the control) and heavy trajectories the norm of the acceleration of the state (or the velocity of the control). Macrosystems arising in social and economic sciences or biological sciences seem to exhibit heavy trajectories.

We make explicit the differential equations providing slow and heavy trajectories when the viability domain is smooth.