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Lyapunov analysis is used to study either the passive dynamics of a system or the dynamics of a closed-loop system (e.g. the plant and controller connected in a feedback loop). We will see generalizations of the Lyapunov function ideas to input-output systems later in the text.

(Lyapunov stability theorem): If x = 0 is an equilibrium point of system (2.31) and define positive scalar function V(x) near the equilibrium point U0 ⊂ U with continuous derivative V ̇ x , then we have (1) for any ∀ x ∈ U0 if V ̇ x ≤ 0 is satisfied, system is stable at x = 0; (2) for any ∀ x ∈ U0 if V ̇

The (continuous) Lyapunov equation is AX + XAH + Q = 0 where Q is hermitian. This is a special case of the Sylvester equation.

The Lyapunov approach is based on the physical idea that the energy of an isolated system decreases. A Lyapunov function maps scalar or vector variables to real numbers (ℜN → ℜ+) and decreases with time.

(Lyapunov stability theorem): If x = 0 is an equilibrium point of system (2.31) and define positive scalar function V(x) near the equilibrium point U0 ⊂ U with continuous derivative V ̇ x , then we have (1) for any ∀ x ∈ U0 if V ̇ x ≤ 0 is satisfied, system is stable at x = 0; (2) for any ∀ x ∈ U0 if V ̇

The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems.

An Example of Lyapunov Functions The function V(x, y) is obviously positive everywhere, except at the origin where it is zero. Now, let's calculate the total derivative of the function V(x, y). dV/dt = -16[(x/2) – y]2 < 0.

Control and Physical Intelligence and suppose that f has an equilibrium at xe such that f(xe) = 0. Lyapunov stability [15] of an equilibrium means that solutions starting in vicinity to the equilibrium (within a distance δ from it) remain in vicinity forever (within a distance ϵ from it).

Theorem 1.55 (Lyapunov's Stability Theorem). If V is a strict Lyapunov function, then the solution curve definitely crosses the level set Sc and remains inside the set {x ∈ Rn : V (x) < c} for all t > 0. Because the same property holds at all level sets “inside” Sc, the rest point x0 is asymptotically stable.