How to Find a Lyapunov Function for a Nonlinear Time-Varying ODE

A Lyapunov function is a way to study the stability of an equilibrium without solving the differential equation explicitly. For a non-autonomous nonlinear system, the right-hand side depends on time as well as the state, so the Lyapunov function may depend on both.

Write the system as

$$\dot{x}=f(t,x), \qquad x\in\mathbb{R}^n.$$

The usual target is the equilibrium $x=0$. It must actually be an equilibrium: $f(t,0)=0$ for every time under consideration. The key test is to find a function that is positive near the equilibrium and decreases along trajectories.

Differentiate the function along a solution

Let $V(t,x)$ be continuously differentiable. Along any solution $x(t)$, the chain rule gives

$$\dot V(t,x)=\frac{\partial V}{\partial t}(t,x)+\nabla_x V(t,x)\,f(t,x).$$

The term $\partial V/\partial t$ is essential if $V$ depends explicitly on time. Omitting it can lead to a wrong stability conclusion. If $V$ depends only on $x$, that term is zero, and the formula reduces to the familiar autonomous-system expression.

A useful stability test compares $V$ with the distance from the equilibrium. For example, suppose in a region around the origin there are class $\mathcal{K}$ functions $\alpha_1$ and $\alpha_2$ (continuous, strictly increasing, and zero at zero) such that

$$\alpha_1(\lVert x\rVert)\le V(t,x)\le\alpha_2(\lVert x\rVert).$$

These uniform-in-time bounds say that $V$ is positive away from the equilibrium and does not become arbitrarily small or large merely because time changes. If also $\dot V\le 0$, the equilibrium is stable under the standard Lyapunov conditions. If, more strongly,

$$\dot V(t,x)\le-\alpha_3(\lVert x\rVert),$$

for another class $\mathcal{K}$ function $\alpha_3$, trajectories are driven toward the equilibrium; with the stated uniform bounds, this gives uniform asymptotic stability. If the bounds and derivative inequality hold globally, and the lower bound is radially unbounded (for example, $\alpha_1$ is class $\mathcal{K}_\infty$), the strict derivative bound gives global asymptotic stability, provided solutions exist for all forward time. With only $\dot V\le 0$, the global conclusion is stability, not necessarily convergence. Local inequalities establish only local results.

A worked nonlinear, time-varying example

Consider the scalar equation

$$\dot{x}=-(1+\sin^2 t)x-x^3.$$

The coefficient varies with time, and the cubic term makes the system nonlinear. The origin is an equilibrium because the right-hand side is zero at $x=0$ for every $t$.

Try the simple candidate

$$V(x)=\frac{1}{2}x^2.$$

It is positive for every $x\ne0$ and equals zero at the equilibrium. Since $V$ has no explicit time dependence, its derivative along a solution is

$$\begin{aligned} \dot V &=x\dot{x}\\ &=x\bigl(-(1+\sin^2 t)x-x^3\bigr)\\ &=-(1+\sin^2 t)x^2-x^4\\ &\le -x^2=-2V. \end{aligned}$$

The inequality uses $\sin^2 t\ge0$ and $x^4\ge0$. Therefore $V$ decreases at least as fast as the solution of $\dot V=-2V$. Integrating that comparison gives

$$V(t)\le V(t_0)e^{-2(t-t_0)}, \qquad t\ge t_0.$$

Because $V=x^2/2$, this implies

$$|x(t)|\le |x(t_0)|e^{-(t-t_0)}.$$

Thus the origin is globally exponentially stable: every initial value converges to zero, with an explicit exponential bound. This example also shows that a time-varying system does not necessarily require a time-varying Lyapunov function.

How to choose a candidate

There is no universal recipe, but these starting points are common:

• For systems with a linearization that suggests a quadratic energy, try $V(x)=x^TPx$, where $P$ is symmetric positive definite.
• For scalar equations, try $V(x)=x^2/2$ or another positive power of $|x|$ that is differentiable where needed.
• For mechanical or physical models, try an energy-like expression, such as kinetic plus potential energy.
• If the coefficients or forcing vary with time, consider $V(t,x)$, or a state-only candidate whose derivative can still be bounded uniformly in time.

A candidate need not make the derivative strictly negative everywhere to prove stability. A nonpositive derivative can be enough for stability, while asymptotic convergence generally requires additional conditions, such as a negative-definite bound or a suitable invariance argument. For non-autonomous systems, the exact theorem and its uniformity assumptions matter; checking only that $V$ decreases at each instant may not establish uniform convergence.

Common mistakes

• Ignoring explicit time dependence: For $V(t,x)$, include $\partial V/\partial t$ in the derivative.
• Checking positivity only at one time: A time-dependent candidate should satisfy the needed positive bounds throughout the time interval, typically uniformly in time.
• Confusing stability with convergence: A nonincreasing $V$ can keep trajectories near the equilibrium without proving they approach it.
• Claiming a global result from local bounds: If the inequalities hold only near zero, the conclusion is local.
• Forgetting existence of solutions: A global stability claim also requires that solutions continue to exist for all future time.

The practical method is to verify the equilibrium, choose a positive candidate, compute its full derivative along trajectories, and match the resulting inequalities to a stability theorem whose assumptions apply to the region and time interval you are studying.