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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.
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 .
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.
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.
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 ...
If all eigenvalues of the matrix have negative real parts, and X(x, y), Y(x, y) vanish when x = 0, then the solution x = 0, y = 0 of this system is stable with respect to (x, y) and asymptotically stable with respect to x. If a solution (x(t), y(t)) is close enough to the solution x = 0, y = 0, then
The general result was independently discovered by J.P. LaSalle (then at RIAS) and N.N. Krasovskii, who published in 1960 and 1959 respectively. While LaSalle was the first author in the West to publish the general theorem in 1960, a special case of the theorem was communicated in 1952 by Barbashin and Krasovskii , followed by a publication of ...