Internal Stability

Internal stability deals with the zero-input response ($\dot{x}=Ax$). Does the system converge towards its equilibrium point (or in a neighborhood around it)? Think a pendelum and an inverted pendelum

Definitions

The dynamics of the system can be written as

$$\dot{x}=Ax\to x(t)=e^{At}{x}_0$$

For example: If $A$ is the zero matrix, we get that $\dot{x}=x_0$ meaning that the system is marginally stable