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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
A Lyapunov function for an autonomous dynamical system {: ˙ = ()with an equilibrium point at = is a scalar function: that is continuous, has continuous first derivatives, is strictly positive for , and for which the time derivative ˙ = is non positive (these conditions are required on some region containing the origin).
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.
A Lyapunov function is a nonnegative scalar measure of this multi-dimensional state. Typically, the function is defined to grow large when the system moves towards undesirable states. System stability is achieved by taking control actions that make the Lyapunov function drift in the negative direction towards zero.
In stability theory and nonlinear control, Massera's lemma, named after José Luis Massera, deals with the construction of the Lyapunov function to prove the stability of a dynamical system. [1] The lemma appears in (Massera 1949, p. 716) as the first lemma in section 12, and in more general form in (Massera 1956, p. 195) as lemma 2. In 2004 ...
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
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