Control-Lyapunov function

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In control theory, a control-Lyapunov function $V (x, u)$ is a generalization of the notion of Lyapunov function $V (x)$ 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 $x \ne 0$ will eventually return to $x = 0$ . 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 $u (x, t)$ 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

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

where the state x(t) and the control u(t) are vectors.

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

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

The last condition 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 control-Lyapunov function if and only if there exists a regular stabilizing feedback u(x).

It may not be easy to find a control-Lyapunov function 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

$u^*(x) = \arg\min_u \nabla V(x,u) \cdot f(x,u)$