Lyapunov stability

Lyapunov stability means that every chosen neighborhood of an equilibrium contains a smaller neighborhood whose trajectories remain inside the chosen one for all future time.

Lyapunov function: A scalar function whose change along system trajectories can certify stability or convergence. Its value acts like an energy measure that bounds how far trajectories can move.
Dynamical system: A mathematical model specifying how a system's state evolves over time. Lyapunov stability is a property of trajectories generated by such a model.
Instability: Failure of stability, where some arbitrarily small perturbations eventually leave a fixed neighborhood. It is the direct opposite of Lyapunov stability at an equilibrium.
Feedback control: A control method that adjusts inputs using measurements of a system's current state or output. Feedback laws are often designed to make an equilibrium Lyapunov stable.
LaSalle's invariance principle: A theorem that identifies limiting trajectories using a nonincreasing function and its largest invariant set. It strengthens Lyapunov arguments when the candidate function has zero derivative on more than the equilibrium.
Equilibrium point: A state that remains unchanged under a dynamical system's evolution. Stability asks whether trajectories starting nearby remain close to this state.
Exponential stability: Stability in which deviations from an equilibrium decay at least as fast as an exponential bound. It supplies a quantitative convergence rate absent from Lyapunov stability.
Nonlinear control: The analysis and design of controllers for systems with nonlinear dynamics. Lyapunov methods are a central tool when linear control tests do not apply.
Linearization stability theorem: A theorem relating a nonlinear equilibrium's local stability to eigenvalues of its linearization under suitable conditions. It gives a local test when a Lyapunov function is difficult to construct.
State space: The set of all possible states of a system, often represented as a geometric space. Neighborhoods in this space give the precise meaning of perturbations and closeness.

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