Stability theory

Stability theory studies whether solutions of equations remain close to a reference solution after small changes in initial conditions, parameters, or disturbances.

Lyapunov stability: A solution is Lyapunov stable if sufficiently small initial perturbations remain small for all future time. This is the central definition of stability for trajectories of dynamical systems.
Dynamical system: A dynamical system is a rule describing how a state changes over time. Stability theory asks how its trajectories respond to disturbances.
Structural stability: A system is structurally stable if small changes to its defining equations preserve its qualitative behavior. It concerns changes to the system itself, rather than only changes to initial data.
Control theory: Control theory studies how to influence the behavior of systems using inputs and feedback. Controllers are commonly designed to make desired states stable despite disturbances.
Aleksandr Lyapunov: Aleksandr Lyapunov was a Russian mathematician whose work established foundational methods for stability analysis. His 1892 work introduced the function-based approach that bears his name.
Asymptotic stability: A stable equilibrium is asymptotically stable when nearby trajectories also converge to it over time. It strengthens remaining close by requiring disturbances to die away.
Initial value problem: An initial value problem specifies a differential equation together with the state at an initial time. Changing that starting state is one basic test of solution stability.
Numerical stability: Numerical stability describes whether errors in a computational method remain controlled as a calculation proceeds. It concerns discretized computations, not solely the behavior of exact solutions.
Routh–Hurwitz criterion: The Routh–Hurwitz criterion determines whether a polynomial's roots have negative real parts using its coefficients. It tests continuous-time linear stability without calculating every eigenvalue.
Henri Poincaré: Henri Poincaré was a French mathematician who developed qualitative methods for studying differential equations and dynamical systems. His work helped shift attention from explicit formulas to the qualitative behavior of trajectories.

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