Control-Lyapunov function

In control theory a control-Lyapunov function ${\displaystyle V(x,u)}$ is a generalization of the notion of Lyapunov function ${\displaystyle V(x)}$ used in stability analysis. The ordinary Lyapunov function is used to test whether a dynamical system is stable, that is whether the system started in a state ${\displaystyle x\neq 0}$ will eventually return to ${\displaystyle x=0}$. The control-Lyapunov function is used to test whether a system is feedback stabilizable, that is whether for any state x there exists a control ${\displaystyle u(x)}$ such that the system can be brought to the zero state by applying the control u.

The theory and application of CLF's were developed by Z. Artstein and E. D. Sontag in the 1980's.