Control-Lyapunov function

In control theory, a control-Lyapunov function is a Lyapunov function V ( x ) {displaystyle V(x)} for a system with control inputs. The ordinary Lyapunov function is used to test whether a dynamical system is stable (more restrictively, asymptotically stable). That is, whether the system starting in a state x ≠ 0 {displaystyle x eq 0} in some domain D will remain in D, or for asymptotic stability will eventually return to x = 0 {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 u ( x , t ) {displaystyle u(x,t)} such that the system can be brought to the zero state by applying the control u. In control theory, a control-Lyapunov function is a Lyapunov function V ( x ) {displaystyle V(x)} for a system with control inputs. The ordinary Lyapunov function is used to test whether a dynamical system is stable (more restrictively, asymptotically stable). That is, whether the system starting in a state x ≠ 0 {displaystyle x eq 0} in some domain D will remain in D, or for asymptotic stability will eventually return to x = 0 {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 u ( x , t ) {displaystyle u(x,t)} such that the system can be brought to the zero state by applying the control u.