Lyapunov function

A Lyapunov function $V:D\to \mathbb{R}$ is a scalar-valued function on a domain $D$ around an equilibrium that behaves like an energy: it has a minimum at the equilibrium and (typically) decreases along trajectories of the system. Used in Lyapunov’s method to determine stability without solving the ODE.

For an autonomous system ${\mathbf{x}}^{\prime }=\mathbf{F}(\mathbf{x})$ with equilibrium at $0$, a candidate Lyapunov function $V$ should satisfy:

1. $V(0)=0$ (zero energy at the equilibrium).
2. $V(\mathbf{x})>0$ for $\mathbf{x}\in D\setminus \{0\}$ (positive energy elsewhere — positive definite).
3. $\dot{V}(\mathbf{x})\le 0$ along trajectories (energy never increases).

If $\dot{V}$ is strictly negative (negative definite) on $D\setminus \{0\}$, the equilibrium is asymptotically stable. If $\dot{V}\le 0$ (negative semi-definite), the equilibrium is at least stable.

What “behaves like energy” means

Energy in a conservative mechanical system (no friction, no driving) is conserved: $\dot{E}=0$. Solutions trace level sets of $E$ — closed orbits at constant energy. The equilibrium has minimum energy.

In a dissipative system (with friction), energy decreases: $\dot{E}<0$. Solutions move from higher- to lower-energy states, ultimately settling at the equilibrium of minimum energy.

Lyapunov functions generalize this idea. Any $V$ with the right properties (positive definite + non-increasing along trajectories) plays the same role as energy, and the conclusions about stability follow.

Computing $\dot{V}$

The time derivative along trajectories:

$\dot{V}(\mathbf{x})=\frac{d}{dt}V(\mathbf{x}(t))=\nabla V(\mathbf{x})\cdot {\mathbf{x}}^{\prime }(t)=\nabla V(\mathbf{x})\cdot \mathbf{F}(\mathbf{x})$

You don’t need to know the explicit solution $\mathbf{x}(t)$ — just compute the gradient of $V$ and the dot product with the right-hand side.

For 2D: $\dot{V}=V_x\cdot f+V_y\cdot g$ where $f,g$ are the components of $\mathbf{F}$.

Definitions

A scalar function $V$ on $D$ around $0$ is:

• Positive definite: $V(0)=0$ and $V(\mathbf{x})>0$ for $\mathbf{x}\ne 0$.
• Positive semi-definite: $V(0)=0$ and $V(\mathbf{x})\ge 0$.
• Negative definite: $-V$ is positive definite.
• Negative semi-definite: $-V$ is positive semi-definite.

For Lyapunov-stability theorems, the requirements are:

• Stable: $V$ positive definite, $\dot{V}$ negative semi-definite.
• Asymptotically stable: $V$ positive definite, $\dot{V}$ negative definite.

Examples of Lyapunov functions

Quadratic forms: $V(\mathbf{x})={\mathbf{x}}^{T}P\mathbf{x}$ for positive definite matrix $P$. Always positive definite. Computing $\dot{V}$ is straightforward for linear systems:

$\dot{V}={\mathbf{x}}^T(A^{T}P+PA)\mathbf{x}$

For a linear stable system, you can solve the Lyapunov equation $A^{T}P+PA=-Q$ for any positive definite $Q$ to find a $P$ that works.

Energy in mechanical systems: total kinetic + potential energy is often a Lyapunov function. For a damped oscillator, $E=\frac{1}{2}m{\dot{x}}^2+\frac{1}{2}k{x}^2$ is positive definite (around the equilibrium $x=\dot{x}=0$), and $\dot{E}=-\gamma {\dot{x}}^2\le 0$ — negative semi-definite. Lyapunov’s theorem gives stability.

Custom polynomials: try $V=a{x}^2+b{y}^2$ or higher-order forms. Adjust coefficients to make $\dot{V}$ negative definite.

Why finding them is hard

There’s no universal algorithm. Lyapunov-function discovery is part of the art of nonlinear analysis. Common strategies:

1. Energy of the physical system, if it has a natural energy concept.
2. Quadratic guesses with adjustable coefficients, then make $\dot{V}$ negative.
3. Lyapunov equation for linear systems.
4. Sum-of-squares methods (advanced) — solve a semidefinite program to find a polynomial $V$.

For a system known to be stable, some Lyapunov function exists (a converse Lyapunov theorem) — but constructively finding it is sometimes impossible without specialized tools.

In context

For the stability theorems that use the Lyapunov function, see Lyapunov’s method. For the linear theory it complements (when linearization works), see Stability of autonomous systems. For the autonomous system framework, see Autonomous system.