Small control property

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

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