LaSalle invariance principle

LaSalle's invariance principle is a generalization of Lyapunov's direct method that extends the analysis of asymptotic stability to systems where the derivative of the Lyapunov function is only negative semi-definite rather than strictly negative. Developed by Joseph LaSalle in 1960, it states that if a system's trajectories remain in a region where the Lyapunov function is constant, then the system must converge to the largest invariant set within that region. This largest invariant set may be smaller than the entire set of points where the derivative vanishes, allowing stronger conclusions about asymptotic behavior than Lyapunov's original theorem.

The principle is essential for systems with conservation laws or symmetries, where the Lyapunov function naturally ceases to decrease along certain directions. Rather than proving that nothing happens, LaSalle's principle proves that what happens is confined to a specific invariant subset — often the equilibrium itself. In control theory, it is used to prove convergence of adaptive systems and observer designs where the standard Lyapunov conditions fail. The principle reveals that stability is not about universal decrease but about the geometry of invariant sets: a system is stable not because it always descends, but because the only flat places it can reach are the places it wants to be.