Small control property

For applied mathematics, in nonlinear control theory, a non-linear system of the form $\dot{x}=f(x,u)$ is said to satisfy the small control property if for every $\epsilon >0$ there exists a $\delta >0$ so that for all $\Vert x\Vert <\delta$ there exists a $\Vert u\Vert <\epsilon$ 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.

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