Global and point controllability of uncertain dynamical systems

This paper analyzes the problem of controllability in the presence of additive uncertainty. Unlike the stochastic controllability problem, the uncertain parameters here are described in a set-theoretic manner, i.e., no a priori statistics are assumed for the uncertainty
q(\cdot)
. Only a bounding set
Q
is taken as given. Loosely speaking, the state
x(\cdot)
of a dynamical system (
S
) is said to be (
\Omega, Q
)-controllable to a given target
X
if one can guarantee, by choice of a measurable control
u(t)\in \Omega
, the transfer of
x(\cdot)
to
X
in finite time. This "guarantee" above must hold for all measurable
q(t)\in Q
. Known results on constrained controllability of deterministic systems are derived as a special case by setting
Q = {0}
. The finite-dimensional nature of the (
\Omega, Q
)-controllability criteria is perhaps the most salient feature of the results given here. Instead of searching over the function space of controls, one can decide on the question of controllability by solving an "equivalent" problem in
R
" (where
n
is the state dimension). The generation of this equivalent problem involves artificially inducing a saddle point via a certain enveloping operation. A number of examples are given to illustrate the method of computing numerical solutions.