Lyapunov Stability Analysis

Origin. Aleksandr Lyapunov's doctoral thesis (1892); foundational to Soviet control theory; extended by Lur'e, Yakubovich, and the absolute stability school.

Mechanism. Proves stability without solving the system's equations. Construct a Lyapunov function V(x) — a scalar function that is positive definite and decreases along system trajectories. If such a function exists, the system is stable: the state moves toward the equilibrium because V is always decreasing. The method converts a dynamic question (will the trajectory converge?) into a static question (does a suitable V exist?).

Procedure. Identify the equilibrium point. Propose a candidate Lyapunov function — often quadratic, V(x) = x'Px for positive definite P. Compute the time derivative dV/dt along the system trajectories. If dV/dt is negative definite, the equilibrium is asymptotically stable. If only negative semi-definite, apply LaSalle's invariance principle. If no V can be found, the method is inconclusive (not proof of instability).

Applies to. Control system design, verifying stability of nonlinear systems, certifying safety of dynamical systems.

Limitations. Finding a Lyapunov function is an art, not an algorithm (though LMI methods now partially automate it for certain system classes). Failure to find V does not prove instability. The method also gives only stability, not performance; a stable system can still be sluggish or oscillatory within bounds.