A necessary and sufficient condition for local constrained controllability of a linear system

The linear state equation
\dot{x}(t)= A(t)x(t) + B(t)u(t)
is said to be locally Ω-null controllabe if, for every initial condition x
0
in some neighborhood of the origin, there exists a measurable control
u(t)\in\Omega
which steers x
0
to zero in finite time. The set Ω above is prespecified and corresponds to "actuator constraints" which depend on the underlying physical problem. This paper generalizes the known result of [1], i.e, our necessary and sufficient condition for local Ω-null controllability not only holds for the time-invariant systems considered in [1], but also holds for time-varying systems. The local controllability criteria given here complement a number of results given in [6], [7], [9]-[12] on global controllability.