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Lyapunov, in his original 1892 work, proposed two methods for demonstrating stability. [1] The first method developed the solution in a series which was then proved convergent within limits. The second method, which is now referred to as the Lyapunov stability criterion or the Direct Method, makes use of a Lyapunov function V(x) which has an ...
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.
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.
If, in addition, all eigenvalues of have negative real parts (is stable), and the unique solution of the Lyapunov equation + = is positive definite, the system is controllable. The solution is called the Controllability Gramian and can be expressed as W c = ∫ 0 ∞ e A τ B B T e A T τ d τ {\displaystyle {\boldsymbol {W_{c}}}=\int _{0 ...
In particular, the discrete-time Lyapunov equation (also known as Stein equation) for is A X A H − X + Q = 0 {\displaystyle AXA^{H}-X+Q=0} where Q {\displaystyle Q} is a Hermitian matrix and A H {\displaystyle A^{H}} is the conjugate transpose of A {\displaystyle A} , while the continuous-time Lyapunov equation is
Lyapunov proved that if the system of the first approximation is regular (e.g., all systems with constant and periodic coefficients are regular) and its largest Lyapunov exponent is negative, then the solution of the original system is asymptotically Lyapunov stable. Later, it was stated by O. Perron that the requirement of regularity of the ...
Lyapunov theory, a theorem related to the stability of solutions of differential equations near a point of equilibrium; Lyapunov central limit theorem, variant of the central limit theorem; Lyapunov vector-measure theorem, theorem in measure theory that the range of any real-valued, non-atomic vector measure is compact and convex