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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 analogy to the potential
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).
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 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
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 ...
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–Schmidt reduction has been used in economics, natural sciences, and engineering [1] often in combination with bifurcation theory, perturbation theory, and regularization. [ 1 ] [ 2 ] [ 3 ] LS reduction is often used to rigorously regularize partial differential equation models in chemical engineering resulting in models that are ...
In 1930 O. Perron constructed an example of a second-order system, where the first approximation has negative Lyapunov exponents along a zero solution of the original system but, at the same time, this zero solution of the original nonlinear system is Lyapunov unstable. Furthermore, in a certain neighborhood of this zero solution almost all ...