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Lyapunov Function
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. A similar concept appears in the theory of general state space Markov chains, usually under the name Foster–Lyapunov functions. For certain classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known. For instance, quadratic functions suffice for systems with one state; the solution of a particular linear matrix inequality provides Lyapunov functions for linear systems; and conservation laws can often be used to construct Lyapunov functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equations
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive-definite Function
In mathematics, a positive-definite function is, depending on the context, either of two types of function. Most common usage A ''positive-definite function'' of a real variable ''x'' is a complex-valued function f: \mathbb \to \mathbb such that for any real numbers ''x''1, …, ''x''''n'' the ''n'' × ''n'' matrix : A = \left(a_\right)_^n~, \quad a_ = f(x_i - x_j) is positive ''semi-''definite (which requires ''A'' to be Hermitian; therefore ''f''(−''x'') is the complex conjugate of ''f''(''x'')). In particular, it is necessary (but not sufficient) that : f(0) \geq 0~, \quad , f(x), \leq f(0) (these inequalities follow from the condition for ''n'' = 1, 2.) A function is ''negative semi-definite'' if the inequality is reversed. A function is ''definite'' if the weak inequality is replaced with a strong ( 0). Examples If (X, \langle \cdot, \cdot \rangle) is a real inner product space, then g_y \colon X \to \mathbb, x \mapsto \exp(i \langle y, x \r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foster's Theorem
In probability theory, Foster's theorem, named after Gordon Foster, is used to draw conclusions about the positive recurrence of Markov chains with countable state spaces. It uses the fact that positive recurrent Markov chains exhibit a notion of "Lyapunov stability" in terms of returning to any state while starting from it within a finite time interval. Theorem Consider an irreducible discrete-time Markov chain on a countable state space ''S'' having a transition probability matrix P with elements ''p''''ij'' for pairs ''i'', ''j'' in ''S''. Foster's theorem states that the Markov chain is positive recurrent if and only if there exists a Lyapunov function V: S \to \mathbb, such that V(i) \geq 0 \text \forall \text i \in S and # \sum_p_V(j) < for # for all for some finite set ''F'' and strictly positive ''ε''. Related links * |
Chetaev Function
The Chetaev instability theorem for dynamical systems states that if there exists, for the system \dot = X(\textbf) with an equilibrium point at the origin, a continuously differentiable function V(x) such that # the origin is a boundary point of the set G = \; # there exists a neighborhood U of the origin such that \dot(\textbf)>0 for all \mathbf \in G \cap U then the origin is an unstable equilibrium point of the system. This theorem is somewhat less restrictive than the Lyapunov instability theorems, since a complete sphere (circle) around the origin for which V and \dot both are of the same sign does not have to be produced. It is named after Nicolai Gurevich Chetaev. Applications Chetaev instability theorem has been used to analyze the unfolding dynamics of proteins under the effect of optical tweezers. See also * Lyapunov function — a function whose existence guarantees stability References * Further reading *{{cite journal , doi=10.4249/scholarpedia.4672, doi-ac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Control-Lyapunov Function
In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V(x) to systems with control inputs. 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 \ne 0 in some domain ''D'', then the state will remain in ''D'' for all time. For ''asymptotic stability'', the state is also required to converge to x = 0. A control-Lyapunov function is used to test whether a system is ''asymptotically stabilizable'', that is whether for any state ''x'' there exists a control u(x,t) such that the system can be brought to the zero state asymptotically by applying the control ''u''. The theory and application of control-Lyapunov functions were developed by Zvi Artstein and Eduardo D. Sontag in the 1980s and 1990s. Definition Consider an autonomous dynamical system with inputs where x\in\mathbb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathem ... |