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The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but "nearby" solutions to differential equations. Input-to-state stability (ISS) applies Lyapunov notions to systems with inputs.
The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems. [ 1 ] [ 2 ] In particular, the discrete-time Lyapunov equation (also known as Stein equation ) for X {\displaystyle X} is
The ordinary Lyapunov function is used to test whether a dynamical system is (Lyapunov) stable or (more restrictively) asymptotically stable. Lyapunov stability means that if the system starts in a state x ≠ 0 {\displaystyle x\neq 0} in some domain D , then the state will remain in D for all time.
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory .
Lyapunov functions are used extensively in control theory to ensure different forms of system stability. The state of a system at a particular time is often described by a multi-dimensional vector. A Lyapunov function is a nonnegative scalar measure of this multi-dimensional state.
ISS unified the Lyapunov and input-output stability theories and revolutionized our view on stabilization of nonlinear systems, design of robust nonlinear observers, stability of nonlinear interconnected control systems, nonlinear detectability theory, and supervisory adaptive control. This made ISS the dominating stability paradigm in ...
Comparison functions are primarily used to obtain quantitative restatements of stability properties as Lyapunov stability, uniform asymptotic stability, etc. These restatements are often more useful than the qualitative definitions of stability properties given in ε - δ {\displaystyle \varepsilon {\text{-}}\delta } language.
The Lyapunov–Malkin theorem (named for Aleksandr Lyapunov and Ioel Malkin ) is a mathematical theorem detailing stability of nonlinear systems. [ 1 ] [ 2 ] Theorem