The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems. [1][2] In particular, the discrete-time Lyapunov equation (also known as Stein equation) for is.

The Lyapunov equation the Lyapunov equation is ATP +PA+Q = 0 where A, P, Q ∈ Rn×n, and P, Q are symmetric interpretation: for linear system x˙ = Ax, if V(z) = zTPz, then V˙ (z) = (Az)TPz +zTP(Az) = −zTQz i.e., if zTPz is the (generalized)energy, then zTQz is the associated (generalized) dissipation

Lyapunov theory is used to make conclusions about trajectories of a system x˙ = f(x) (e.g., G.A.S.) without finding the trajectories (i.e., solving the differential equation)

Lyapunov equation is used in estimating the rates at which x 0. Lyapunov function is used to analyze Lyapunov controllers, observers , etc. Our interest in Lyapunov equation stems from control and filtering applications rather than stability. What does J(u) mean? "We want x 0 from x without using too much control."

Lyapunov equation. If A is stable and Q 0, then P 0. If A is stable, Q 0, and (Q;A) observable, then P > 0. Lyapunov equation solvability conditions The discrete-time Lyapunov equation has a unique solution P, for any Q = QT, if and only if i(A) j(A) 6= 1, for i;j = 1;:::;n.

Both the Lyapunov’s indirect method (Theorem L.5) and direct method (Theorem L.1) can be used to judge the local stability of an equilibrium point when the linearized system matrix A is either asymptotically stable or unstable.

Idea of the proof: Sufficiency follows from Lyapunov’s theorem. Necessity Z is shown by verifying that. Suppose A is Hurwitz. Choose Q = QT > 0 and solve the Lyapunov equation P A + AT P = −Q for P . Use. ̇x = ̇V (x) = xT P f (x) + f T (x)P x. ̇V (x) < −[λ min(Q) − …