M. Quincampoix & V. Veliov
Titre:
"Viability under uncertain initial state"
Résumé:
We consider the problem of viable control of a nonlinear system in
$\Rn$ in the case of a non-exactly known initial state. We characterize
the family of those initial sets (and in this sense -- the uncertain
initial states) for which the problem is solvable. The characterization
employs the notion of contingent cone to a given collection of
sets and involves an appropriate set-dynamic equation that describes
the evolution of the state estimation within a prescribed collection
of sets. An extension of the classical concept of viability kernel
with respect to this set-dynamic equation is the key tool. We present
an approximation scheme for the viability kernel, which is numerically
realizable in case of low dimension and simple collections of sets
chosen for state estimation (balls, ellipsoids, polyhedrons, etc.)
As an application we characterize the optimal time function for reaching
a given target, in the case of uncertain initial state. As another
application we consider a viability differential game, where
the uncertainty may enter also in the dynamics of the system as
an input which is not known in advance. The control is then sought
as a nonanticipative strategy depending on the uncertain input.
Sommaire