Foundations — Lyapunov stability — Lyapunov function, positive definiteness

Before you can read the parent note Lyapunov stability you must be fluent in a stack of symbols it throws at you. We build each one from nothing, in an order where every symbol is earned before it is used.

0. The very first thing: what is a "state"?

A machine (a spacecraft, a pendulum, a car) at any instant is completely described by a short list of numbers: where it is and how fast each part is moving. That list is the state.

Look at the red dot above: that single point is the entire machine at this instant. As time passes, the red dot slides around and traces a curve.

1. Motion: velocity, the dot notation, and $\mathbf{f}$

The state does not sit still — it moves. To talk about how fast it moves we need velocity.

Now, who decides that arrow? The physics of the machine does. There is a rule: "if you are at state $\mathbf{x}$, your velocity arrow is this." That rule is the function $\mathbf{f}$.

2. The equilibrium: the point $0$ and $\mathbf{f}(0)=0$

3. Measuring distance: the norm $\Vert \mathbf{x}\Vert$

To say "the dot stays close to the target" we need to measure how far the dot is from the origin.

4. The scalar field $V(\mathbf{x})$ — the "height"

The bowl shape has a name — positive definite — which we now unpack symbol by symbol.

5. The gradient $\nabla V$ and the transpose ${}^{\top }$

To say "downhill" precisely we need the direction of steepest uphill — that is the gradient.

6. The orbital derivative $\dot{V}$ and the chain rule

7. One matrix idea you'll meet: ${\mathbf{x}}^{\top }P\mathbf{x}$

The parent's quadratic bowls look like $V={\mathbf{x}}^{\top }P\mathbf{x}$.

Prerequisite map

Every arrow means "you need the left box to understand the right box." The two boxes positive-definite bowl and downhill $\dot{V}$ feed straight into Lyapunov's theorem — that is the whole topic.

Equipment checklist

Cover the right side; can you answer each before reading on?