Hamilton-Jacobi inequalities on a metric space

In some applied models (of flocking or of the crowd control) it is more natural to deal with elements a metric space (as for instance family subsets vector endowed Hausdorff metric) rather than vectors in normed space. We consider an optimal control problem involving so called morphological system whose trajectories are time dependent tubes RN and show that theory Hamilton-Jacobi-Bellman inequalities can be extended this framework.