**M. Quincampoix & V. M. Veliov **

**Titre:**
"*Optimal Control of Uncertain Systems with Incomplete
Information for the Disturbances*"

**Abstract:**
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 nature: one becomes known to
the controller at the current time (we called it observable),
while the other remains unknown.
The problem can be viewed as a differential
game of min-max type where the controller
aims at minimization of the objective function by a strategy
which depends only on the observable disturbance.
Since the state of the system is not exactly known
due to the presence of a 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 sub-optimal 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 %(which depends on sets)
trough 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, that may be
of independent interest.