Stability Definitions

Considers nonlinear time-invariant system

${\dot{x}}=f(x)$ where $f: \R^n \rightarrow \R^n$

Types of stability:

globally asymptotically stable (G.A.S.) if there is only one unique equilibrium point $x_e$ that the system will settle at given enough time:
- $x(t)\rightarrow x_e$ as $t \rightarrow \infty$
locally asymptotically stable (L.A.S.) if there is a local equilibrium point that the system will settle at given enough time:
- there exists $R>0$ such that $||x(0)-x_e||\leq R \implies x(t) \rightarrow x_e$ as $t\rightarrow\infty$
- there may be more than 1 equilibrium point in such a system

Change of coordinates is commonly used so that $x_e = 0$, uses ${\tilde{x}}=x-x_{e}$

note that for a linear system, there is only one equilibrium point of $x_e=0$ is thus both G.A.S. and L.A.S. (they are the same in this context)

There are other variants of stability but establishing stability when $f$ is nonlinear is typically difficult

Energy and Dissipation Functions

Considers nonlinear system $\dot{x} = f(x)$ and a function $V: \R^n \rightarrow \R$

can define: $\dot{V} : \R^n \rightarrow \R$ as $\dot{V}(z) = \nabla V(z)^T f(z)$
thereby, $\dot{V}(z)=\frac{d}{dt}V(x(t))$ when $z=x(t), \dot{x} = f(x)$