Lyapunovfüggvények

Lyapunov functions are a fundamental concept in the stability analysis of dynamical systems. Introduced by the Russian mathematician Aleksandr Lyapunov, they provide a way to determine the stability of an equilibrium point of a system without explicitly solving its differential equations. A Lyapunov function is a scalar function of the system's state variables that behaves in a specific way around the equilibrium point.

For a system described by the differential equation dx/dt = f(x), where x is the state vector and f(0) = 0 (meaning the origin is an equilibrium point), a Lyapunov function V(x) must satisfy certain conditions. First, V(0) must be zero, and V(x) must be positive for all other states x near the origin. This ensures that the function has a minimum at the equilibrium point. Second, the time derivative of V(x) along the trajectories of the system, denoted as dV/dt, must be negative (or non-positive) for all states x near the origin, except at the origin itself.

If such a function V(x) can be found, it implies that the system is stable. The intuition is that V(x) acts like an "energy" function. If the system starts near the equilibrium, its "energy" must decrease over time, meaning it must move towards the equilibrium point. If the time derivative dV/dt is strictly negative, the equilibrium is asymptotically stable, meaning not only is it stable, but trajectories starting near it will converge to the equilibrium point as time goes to infinity. The existence of a Lyapunov function is a sufficient condition for stability, but not always a necessary one.