For

applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemat ...

, in

nonlinear control Nonlinear control theory is the area of control theory which deals with systems that are nonlinear, time-variant, or both. Control theory is an interdisciplinary branch of engineering and mathematics that is concerned with the behavior of d ...

theory, a

non-linear system In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...

of the form

\dot = 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 In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s se ...

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 Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...

of the system can be found that is asymptotically stabilizable by such an input.

References

Nonlinear control {{systemstheory-stub