I was glad to gain insights from Prof. Eric Feron’s Non-Linear Control course, where I learned about two important theorems that state the following.

Definitions

Stability

Asymptotic Stability. The system \(x(t)\) is asymptotic stable if \(\lim_{t \to \infty} x(t) = 0.\)

Exponential Stability. The system \(x(t)\) is exponential stable if \(\|x(t)\| \le c\cdot \exp(-\alpha t)\|x_0\|.\)

An Invariant Set. An invariant set is a set for a dynamical system if, whenever the system starts be in the set it remains in the set for all future time.

Lyapunov Function

Assume that we can define a function \(V(x): \mathbb{R}^d \to \mathbb{R}^+\), and this function is continuously differentiable. Next, \(V(x) = 0\) if and only if \(x = 0\). Finally, \(V(x)\) is radially unbounded in the sense that for all \(M \in \mathbb{R}\), there exists \(R \in \mathbb{R}\) such that

\[\|x\|^2 \geq R^2 \implies V(x) \geq M.\]

Theorems

Lyapunov Theorem (1892)

Lyapunov’s theorem provides a sufficient condition for asymptotic stability for dynamics. We assume that there is a dynamic process

\[\frac{dx}{dt} = f(x), \quad x(0) = x_0\]

driven by an ODE. By the chain rule, we can evaluate the time derivative of the Lyapunov function:

\[\frac{dV(x)}{dt} = \langle \nabla V(x(t)), \frac{dx}{dt} \rangle = \langle \nabla V(x(t)), f(x(t)) \rangle.\]

If

\[\frac{dx}{dt} = f(x), \quad x(0) = x_0\]

and there exists a Lyapunov function \(V(x): \mathbb{R}^d \to \mathbb{R}^+\) such that

\[-\langle \nabla V(x), f(x) \rangle > 0, \quad \forall x(t) \neq 0,\]

then

\[\lim_{t \to \infty} x(t) = 0.\]

The caveat is that finding such a function \(V(x)\) is generally very hard. Additional references on how to construct \(V(x)\) can be found in the works of Yakubovich, Popov and Arkadiy Nemirovski.

LaSalle’s Theorem (1960)

LaSalle’s theorem generalizes Lyapunov theorem and is useful for analyzing systems with more complex behavior.

Consider the process

\[\frac{dx}{dt} = f(x), \quad x(0) = x_0.\]

If the set \(I = \left\{ x : \langle \nabla V(x), f(x) \rangle = 0 \right\}\) is empty except for \(x(t)=0\) then

\[\lim_{t \to \infty} x(t) = 0.\]