Small control property explained

For applied mathematics, in nonlinear control theory, a non-linear system of the form

• x = f(x,u) is said to satisfy the small control property if for every \varepsilon > 0 there exists a \delta > 0 so that for all \|x\| < \delta there exists a \|u\| < \varepsilon 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.