On a Certain Controllability Gap

Abstract

This paper is concerned with the problem of steering the state $x(t)$ of the system $\dot x(t) = Ax(t) + Bu(t)$ to a prescribed closed and convex target set X in $R^n $. 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 [B. R. Barmish and W. E. Schmitendorf, IEEE Trans. Automat. Control, AC-25 (1980), pp. 540–547.] 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. In this paper, we show that the controllability gap vanishes for a large class of systems. Namely if X is compact, the interior of $\Omega $ is nonempty and $[{\operatorname{rank}}B \vdots AB \vdots A^2 B \vdots \cdots A^{n - 1} B] = n$, then the sufficient condition of Barmish and Schmitendorf is also a necessary condition for global $\Omega $-controllability to X. We also give examples to show that the controllability gap persists if these additional assumptions are not made.