Control-Lyapunov function

In <a href="/facts/Control_theory/aIuERpn6">control theory</a>, a control-Lyapunov function (CLF) is an extension of the idea of <a href="/facts/Lyapunov_function/RB590Utb">Lyapunov function</a> 
 
 
 
 V
 (
 x
 )
 
 
 {\displaystyle V(x)}
 
 to <a href="/facts/Control_system/8CEjIDlS">systems with control inputs</a>. The ordinary Lyapunov function is used to test whether a <a href="/facts/Dynamical_system/782gREl5">dynamical system</a> is <a href="/facts/Lyapunov_stability/r3je5zaQ">(Lyapunov) stable</a> or (more restrictively) asymptotically stable. Lyapunov stability means that if the system starts in a state 
 
 
 
 x
 ≠
 0
 
 
 {\displaystyle x\neq 0}
 
 in some domain D, then the state will remain in D for all time. For asymptotic stability, the state is also required to converge to 
 
 
 
 x
 =
 0
 
 
 {\displaystyle x=0}
 
. A control-Lyapunov function is used to test whether a system is <a href="/facts/Controllability/p1TLxgLf">asymptotically stabilizable</a>, that is whether for any state x there exists a control 
 
 
 
 u
 (
 x
 ,
 t
 )
 
 
 {\displaystyle u(x,t)}
 
 such that the system can be brought to the zero state asymptotically by applying the control u.
The theory and application of control-Lyapunov functions were developed by Zvi Artstein and <a href="/facts/Eduardo_D._Sontag/JKV07RHI">Eduardo D. Sontag</a> in the 1980s and 1990s.

Control-Lyapunov function open-in-new

Control-Lyapunov function