# What is Lyapunov stability theorem?

## What is Lyapunov stability theorem?

The Lyapunov stability theory was originally developed by Lyapunov (Liapunov (1892)) in the context of stability of a nonlinear system. For the linear system: x ˙ ( t ) = A x ( t ) , the function V (x) = xTXx, where X is symmetric is a Lyapunov function if the V ˙ ( x ) , the derivative of V(x), is negative definite.

## How is Lyapunov stability calculated?

Theorem on stability in the sense of Lyapunov. If in a neighborhood U of the zero solution X=0 of an autonomous system there is a Lyapunov function V(X), then the equilibrium point X=0 of the system is Lyapunov stable.

For what conditions is this system stable in the sense of Lyapunov?

1. If V (x, t) is locally positive definite and ˙V (x, t) ≤ 0 locally in x and for all t, then the origin of the system is locally stable (in the sense of Lyapunov).

### What are the conditions for asymptotically stable at the origin?

Moreover, if W(x) is positive definite, then, the equilibrium is asymptotically stable. In addition, if D=Rn and V is radially unbounded, i.e., x V x → ∞ ⇒ → ∞ ( ) (L. 7) then, the origin is globally asymptotically stable.

### How do you determine asymptotic stability?

If V (x) is positive definite and (x) is negative semi-definite, then the origin is stable. 2. If V (x) is positive definite and (x) is negative definite, then the origin is asymptotically stable. then is asymptotically stable.

How do you prove asymptotic stability?

It is asymptotically stable if and only if the eigenvalues of A have strictly negative real part. X = (X + Z) = F(X) = F(X + Z). Therefore Z = F(X + Z).

#### What is the difference between local stability and global stability?

Local stability of an equilibrium point means that if you put the system somewhere nearby the point then it will move itself to the equilibrium point in some time. Global stability means that the system will come to the equilibrium point from any possible starting point (i.e., there is no “nearby” condition).

#### What do you mean by asymptotic stability?

Asymptotic stability means that solutions that start close enough not only remain close enough but also eventually converge to the equilibrium. Exponential stability means that solutions not only converge, but in fact converge faster than or at least as fast as a particular known rate .

What is asymptotic stability in control system?

A time-invariant system is asymptotically stable if all the eigenvalues of the system matrix A have negative real parts. If a system is asymptotically stable, it is also BIBO stable. A system is defined to be exponentially stable if the system response decays exponentially towards zero as time approaches infinity.