Control-Lyapunov function: Difference between revisions

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In [[control theory]], a '''control-Lyapunov function''' <math>V(x,u)</math> is a generalization of the notion of [[Lyapunov function]] <math>V(x)</math> used in [[Lyapunov stability|stability]] analysis. The ordinary Lyapunov function is used to test whether a [[dynamical system]] is ''stable'' (more restrictively, that''asymptotically stable''). That is, whether the system startedstarting in a state <math>x \ne 0</math> in some ___domain ''D'' will remain in ''D'', or for ''asymptotic stability'' will eventually return to <math>x = 0</math>. 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 <math> u(x,t)</math> 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