Small control property

In nonlinear control theory, a non-linear system of the form [math]\displaystyle{ \dot{x} = f(x,u) }[/math] is said to satisfy the small control property if for every [math]\displaystyle{ \varepsilon \gt 0 }[/math] there exists a [math]\displaystyle{ \delta \gt 0 }[/math] so that for all [math]\displaystyle{ \|x\| \lt \delta }[/math] there exists a [math]\displaystyle{ \|u\| \lt \varepsilon }[/math] so that the time derivative of the system's Lyapunov function is negative definite at that point.

In other words, even if the control input is arbitrarily small, a starting configuration close enough to the origin of the system can be found that is asymptotically stabilizable by such an input.