Sub-Optimal Control in Presence of Unobservable Disturbances

We investigate the problem of optimization of a terminal cost function for a system depending on a control, and on two disturbances for which a priori set membership is known. The disturbances are of different natures: One becomes known to the controller at the current time (we called it observable) while the other remains unknown. The state of the system is not measured, therefore not exactly known due to the presence of an unobservable disturbance. We reformulate the problem through a set-valued dynamics describing the evolution of the current set estimation of the state. To reduce the complexity of the problem, we pass to a suboptimal problem where the evolution of the state estimation is restricted to a prescribed collection of sets. The main result of the paper is a characterization of the value function of this problem through a Hamilton–Jacobi inequality in terms of Dini derivatives, which implies a convergent scheme for numerical computations. As necessary auxiliary tools, we provide new results on evolution and viability of tubes in a given collection of sets.