# Stability of Nonlinear Systems

The equilibria of a system can tell us a great deal about the stability of the system. For nonlinear systems, stability is a property of the equilibrium points, and to be stable is to converge to or stay equilibrium.

Lyapunov Stability essentially says that a finite deviation in the initial condition from equilibrium means the resulting trajectory of the system stay close to equilibrium. Notice that this definition is nearly identical to Theorem 10. That means stability of an equilibrium point is the same as saying the function which returns the solution to a system given its initial condition is continuous at the equilibrium point.

Attractive equilibria guarantee that trajectories beginning from initial conditions inside of a ball will converge to the equilibrium. However, attractivity does not imply stability since the trajectory could go arbitarily far from the equilibrium so long as it eventually returns.

Asymptotic stability fixes the problem of attractivity where trajectories could go far from the equilibrium, and it fixes the problem with stability where the trajectory may not converge to equilibrium. It means that trajectories starting in a ball around equilibrium will converge to equilibrium without leaving that ball. Because the constant for attractivity may depend on time, defining uniform asymptotic stability requires some modifications to the idea of attractivity.

Just as we can define stability, we can also define instability.

## Lyapunov Functions

In order to prove different types of stability, we will construct functions which have particular properties around equilibrium points of the system. The properties of these functions will help determine what type of stable the equilibrium point is.

LPDF functions are locally “energy-like” in the sense that the equilibrium point is assigned the lowest “energy” value, and the larger the deviation from the equilibrium, the higher the value of the “energy”.

Descresence means that for a ball around the equilibrium, we can upper bound the the energy.

### Quadratic Lyapunov Functions

### Sum-of-Squares Lyapunov Functions

is SOS.

## Proving Stability

The results of Theorem 16, Theorem 17, Theorem 18, Theorem 19, Theorem 20 are summarized in Table 1.

### Indirect Method of Lyapunov

The state transition matrix is useful in determining properties of the system.

The Lyapunov Lemma has extensions to the time-varying case.

It turns out that uniform asymptotic stability of the linearization of a system corresponds to uniform, asymptotic stability of the nonlinear system.

## Proving Instability

## Region of Attraction

For asymptotically stable and exponential stable equilibria, it makes sense to wonder which initial conditions will cause trajectories to converge to the equilibrium.

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