ON A CERTAIN CONTROLLABILITY GAP.

The problem of steering the state x(t) of the system dx(t)/dt equals Ax(t) plus Bu(t) to a prescribed closed and convex target set X in R**n is considered. In addition, admissible control values u(t) are constrained to lie in a prescribed compact set OMEGA in R**m. In light of the constraint, the term constrained controllability problem is often used to describe the situation above. More precisely, the system is said to be globally OMEGA -controllable to X if every initial state x//0 can be steered to X in finite time. In a recent paper two separate conditions were given for global OMEGA -controllability to X. The first of these conditions was a necessary condition and the second of these conditions was a sufficient condition. This led to the possibility of a so-called controllability gap; that is, the possible existence of systems which satisfy one condition but not the other. It is shown that the controllability gap vanishes for a large class of systems. Examples are given to show that the controllability gap persists if these additional assumptions are not made.