6.5 LaSalle's invariance principle

Definitions of LaSalle's invariance principle

LaSalle's invariance principle extends Lyapunov stability theory by letting you prove convergence even when your Lyapunov function's derivative is only negative semidefinite (not strictly negative definite). Standard Lyapunov analysis gets stuck in this situation because $\dot{V}(x) \leq 0$ doesn't immediately tell you the system is asymptotically stable. LaSalle's principle resolves this by identifying exactly where trajectories end up: the largest invariant set contained in the region where $\dot{V}(x) = 0$.

Autonomous systems and equilibrium points

The principle applies to autonomous systems, meaning systems whose dynamics don't explicitly depend on time:

$\dot{x} = f(x)$

An equilibrium point $x_e$ is a state where $f(x_e) = 0$, so the system stays put once it arrives there. Common examples include pendulums, electrical circuits, and population dynamics models.

LaSalle's principle is especially useful for assessing stability of these equilibrium points when a strict Lyapunov argument falls short.

Invariant sets and limit sets

• An invariant set is a subset of the state space such that any trajectory starting inside it stays inside it for all future time. Equilibrium points, limit cycles, and attractors are all examples.
• A limit set (specifically, the omega-limit set) is the collection of points a trajectory approaches as $t \to \infty$.

LaSalle's principle connects these two ideas: trajectories converge to the largest invariant set $M$ contained within $\{x \in \Omega \mid \dot{V}(x) = 0\}$. If that largest invariant set turns out to be just the equilibrium point, you've proven asymptotic stability.

Lyapunov functions and stability

A Lyapunov function $V(x)$ is a scalar-valued function that is non-increasing along the system's trajectories. Standard Lyapunov stability requires $\dot{V}(x) < 0$ (strictly negative definite) to conclude asymptotic stability.

LaSalle's principle relaxes this: you only need $\dot{V}(x) \leq 0$ (negative semidefinite). The trade-off is that you then need to do extra work to identify the largest invariant set where $\dot{V}(x) = 0$. Typical Lyapunov function candidates include quadratic forms like $V(x) = x^T P x$ and energy-like functions drawn from the physics of the system.

Conditions for LaSalle's invariance principle

Three conditions must hold simultaneously for the principle to apply. Here they are as a checklist:

1. You have a continuously differentiable Lyapunov function $V(x)$ on a region $\Omega$.
2. The time derivative $\dot{V}(x) \leq 0$ for all $x \in \Omega$ (negative semidefinite).
3. The region $\Omega$ is a compact, positively invariant set.

If all three hold, every trajectory starting in $\Omega$ converges to $M$, the largest invariant set contained in $\{x \in \Omega \mid \dot{V}(x) = 0\}$.

Continuously differentiable Lyapunov functions

$V(x)$ must be $C^1$ (continuously differentiable) throughout $\Omega$. This guarantees that the gradient $\nabla V(x)$ exists everywhere in the region and that the time derivative

$\dot{V}(x) = \nabla V(x) \cdot f(x)$

is well-defined and continuous. Polynomials, exponentials, and trigonometric functions all satisfy this requirement.

Negative semidefinite time derivatives

The condition $\dot{V}(x) \leq 0$ for all $x \in \Omega$ means the Lyapunov function never increases along trajectories. It can stay constant on some trajectories, though. This is the key relaxation compared to standard Lyapunov asymptotic stability, which demands $\dot{V}(x) < 0$ everywhere except the equilibrium.

Because $\dot{V}$ can equal zero on a set larger than just the equilibrium, you need the invariant-set analysis to pin down where trajectories actually settle.

Compact and positively invariant sets

• Compact means closed and bounded. This is needed to guarantee that trajectories have well-defined limit sets (by the Bolzano-Weierstrass property).
• Positively invariant means that if a trajectory starts in $\Omega$, it stays in $\Omega$ for all $t \geq 0$.

A practical way to construct such a set: pick a sublevel set of your Lyapunov function, $\Omega_c = \{x \mid V(x) \leq c\}$. Since $\dot{V} \leq 0$, the value of $V$ can't increase, so trajectories can't leave $\Omega_c$. If $\Omega_c$ is also bounded, it's compact and positively invariant.

Applications of LaSalle's invariance principle

Stability analysis of nonlinear systems

The most common use: proving asymptotic stability when $\dot{V}$ is only negative semidefinite. The procedure is:

1. Propose a Lyapunov function $V(x)$ (often energy-based).
2. Compute $\dot{V}(x)$ and verify $\dot{V}(x) \leq 0$.
3. Find the set $S = \{x \in \Omega \mid \dot{V}(x) = 0\}$.
4. Determine the largest invariant set $M \subseteq S$ by checking which trajectories can stay entirely within $S$.
5. If $M = \{x_e\}$ (only the equilibrium), conclude asymptotic stability.

This approach works even when the system's dynamics are too complex for direct eigenvalue analysis or when $\dot{V}$ stubbornly refuses to be strictly negative definite.

Convergence to invariant sets

Sometimes the largest invariant set $M$ is not just a single point but a richer structure, like a limit cycle or a manifold. LaSalle's principle still tells you that all trajectories in $\Omega$ converge to $M$. This is useful for studying synchronization of coupled oscillators, pattern formation in reaction-diffusion systems, and any scenario where the "steady state" is more complex than a fixed point.

Estimating regions of attraction

The region of attraction is the set of initial conditions from which trajectories converge to the desired equilibrium or invariant set. LaSalle's principle gives you a concrete estimate: any sublevel set $\Omega_c = \{x \mid V(x) \leq c\}$ that is bounded and on which $\dot{V} \leq 0$ is contained in the region of attraction. Larger valid values of $c$ give larger (and more useful) estimates. This is directly relevant to controller design and safety verification.

Relationship to other stability theorems

Comparison with Lyapunov's stability theorem

LaSalle's principle is strictly more general. Whenever Lyapunov's theorem applies, LaSalle's does too (with $M$ turning out to be just the equilibrium). But LaSalle's also handles cases Lyapunov's theorem cannot.

Extensions of Barbashin-Krasovskii theorem

The Barbashin-Krasovskii theorem (sometimes called the global version of Lyapunov's theorem) states that if $V(x)$ is positive definite, radially unbounded, and $\dot{V}(x) \leq 0$, and the only trajectory that can stay in $\{\dot{V} = 0\}$ is the equilibrium, then the equilibrium is globally asymptotically stable. This is essentially LaSalle's principle applied globally with the additional requirement that $V$ is radially unbounded ($V(x) \to \infty$ as $\|x\| \to \infty$). LaSalle's principle generalizes this by not requiring global conditions or convergence to a single equilibrium.

Connections to omega-limit sets

The omega-limit set $\omega(x_0)$ of a trajectory starting at $x_0$ is the set of all accumulation points as $t \to \infty$. LaSalle's principle works by proving that $\omega(x_0) \subseteq M$ for every $x_0 \in \Omega$. The argument relies on two facts:

• $V(x)$ is non-increasing and bounded below on $\Omega$, so it converges to a limit on every trajectory.
• On the omega-limit set, $V$ must be constant, which forces $\dot{V} = 0$ there, placing the omega-limit set inside $S$. Since omega-limit sets are invariant, they must lie inside the largest invariant subset $M \subseteq S$.

This connection to omega-limit sets is the core mechanism behind the proof.

Generalizations and extensions

Non-autonomous systems and time-varying Lyapunov functions

For non-autonomous systems $\dot{x} = f(x, t)$, the standard LaSalle principle doesn't directly apply because the system's vector field changes with time. Extensions exist that use time-varying Lyapunov functions $V(x, t)$ and impose additional uniformity conditions (such as uniform boundedness of trajectories or Barbalat's lemma as a complementary tool). These extensions are more technically involved and require careful treatment of the time dependence.

Discontinuous and non-smooth systems

Systems with discontinuities (sliding mode controllers, mechanical systems with Coulomb friction, power electronic converters) require non-smooth versions of LaSalle's principle. These use tools from non-smooth analysis, such as Clarke's generalized gradient and Filippov solutions, to define what $\dot{V}$ means when $V$ or $f$ isn't differentiable everywhere. The core idea remains the same, but the technical conditions are adapted to handle set-valued derivatives.

Infinite-dimensional systems and PDEs

For systems governed by partial differential equations (heat conduction, wave propagation, fluid dynamics), the state space is infinite-dimensional. LaSalle's principle extends to this setting using functional analysis and semigroup theory. The main additional challenge is establishing the required compactness, since bounded sets in infinite-dimensional spaces are not automatically compact. Precompactness of trajectories (or the existence of a compact absorbing set) becomes a crucial hypothesis.

Examples and case studies

Simple nonlinear systems and phase portraits

Consider a damped pendulum: $\dot{\theta} = \omega$, $\dot{\omega} = -\sin\theta - b\omega$ with damping $b > 0$. Using the energy function $V(\theta, \omega) = 1 - \cos\theta + \frac{1}{2}\omega^2$:

1. Compute $\dot{V} = \omega \sin\theta + \omega(-\sin\theta - b\omega) = -b\omega^2 \leq 0$.

2. The set $S = \{\dot{V} = 0\}$ is $\{\omega = 0\}$.

3. On $\{\omega = 0\}$, the dynamics require $\dot{\omega} = -\sin\theta = 0$, so $\theta = n\pi$.

4. The largest invariant set within a neighborhood of the downward equilibrium is $M = \{(\theta, \omega) = (0, 0)\}$.

5. By LaSalle's principle, the equilibrium $(0, 0)$ is asymptotically stable.

Notice that $\dot{V}$ is only negative semidefinite (it vanishes on the entire $\theta$-axis), so standard Lyapunov analysis alone wouldn't give you asymptotic stability. LaSalle's principle fills that gap.

Control system design and stabilization

LaSalle's principle is widely used to verify that feedback controllers achieve their design goals. For instance, in robotic manipulator control, energy-based Lyapunov functions often yield $\dot{V} \leq 0$ rather than $\dot{V} < 0$. Applying LaSalle's principle confirms that the controlled system converges to the desired configuration. Similar approaches appear in chemical reactor regulation and power system stabilization.

Biological and ecological models

In predator-prey models (Lotka-Volterra type) and epidemic models (SIR type), Lyapunov functions can often be constructed to show $\dot{V} \leq 0$, but strict negativity fails on certain subsets of the state space. LaSalle's principle then determines whether the system converges to an endemic equilibrium, a disease-free equilibrium, or a coexistence state, depending on the structure of the largest invariant set.