Lyapunov function

In the theory of ordinary differential equations (ODEs), Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions (also called the Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general state space Markov chains, usually under the name Foster–Lyapunov functions. In the theory of ordinary differential equations (ODEs), Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions (also called the Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general state space Markov chains, usually under the name Foster–Lyapunov functions. For certain classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known. For instance, quadratic functions suffice for systems with one state; the solution of a particular linear matrix inequality provides Lyapunov functions for linear systems; and conservation laws can often be used to construct Lyapunov functions for physical systems. A Lyapunov function for an autonomous dynamical system with an equilibrium point at y = 0 {displaystyle y=0} is a scalar function V : R n → R {displaystyle V:mathbb {R} ^{n} o mathbb {R} } that is continuous, has continuous first derivatives, is locally positive-definite, and for which − ∇ V ⋅ g {displaystyle - abla {V}cdot g} is also locally positive definite. The condition that − ∇ V ⋅ g {displaystyle - abla {V}cdot g} is locally positive definite is sometimes stated as ∇ V ⋅ g {displaystyle abla {V}cdot g} is locally negative definite. Lyapunov functions arise in the study of equilibrium points of dynamical systems. In R n , {displaystyle mathbb {R} ^{n},} an arbitrary autonomous dynamical system can be written as for some smooth g : R n → R n . {displaystyle g:mathbb {R} ^{n} o mathbb {R} ^{n}.} An equilibrium point is a point y ∗ {displaystyle y^{*}} such that g ( y ∗ ) = 0. {displaystyle g(y^{*})=0.} Given an equilibrium point, y ∗ , {displaystyle y^{*},} there always exists a coordinate transformation x = y − y ∗ , {displaystyle x=y-y^{*},} such that: Thus, in studying equilibrium points, it is sufficient to assume the equilibrium point occurs at 0 {displaystyle 0} . By the chain rule, for any function, H : R n → R , {displaystyle H:mathbb {R} ^{n} o mathbb {R} ,} the time derivative of the function evaluated along a solution of the dynamical system is