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 ${\displaystyle u(x,t)}$ such that the system can be brought to the zero state by applying the control u.

More formally, suppose we are given a dynamical system

${\displaystyle {\dot {x}}(t)=f(x(t))+g(x(t))u(t)}$

Definition. A control-Lyapunov function is a function ${\displaystyle V(x,u)}$ that is continuous, positive-definite (that is V(x,u) is positive except at ${\displaystyle x=0}$ where it is zero), proper (that is ${\displaystyle V(x)\to \infty }$ as ${\displaystyle |x|\to \infty }$), and such that

${\displaystyle \forall x\neq 0,\exists u\qquad {\dot {V}}(x,u)<0}$

(this is the key condition; in words it says that for each state x we can find a control u that will reduce the "energy" V). Intuitively, if in each state we can always find a way to reduce the energy, we should eventually be able to bring the energy to zero, that is to bring the system to a stop. This is made rigorous by the following result:

Artstein's theorem. The dynamical system has a differentiable clf if and only if there exists a regular stabilizing feedback u(x).

It may not be easy to find a clf for a given system, but if we can find one thanks to some ingenuity and luck, then the feedback stabilization problem simplifies considerably, in fact it reduces to solving a static non-linear programming problem

${\displaystyle u^{*}(x)=\arg \min _{u}\nabla V(x,u)\cdot f(x,u)}$

for each state x.

The theory and application of CLF's were developed by Z. Artstein and E. D. Sontag in the 1980's and 1990's.