Continuous-time linear system with only negative real parts
Differential equations
Scope
Fields
Natural sciences
Engineering
Astronomy
Physics
Chemistry
Biology
Geology
Applied mathematics
Continuum mechanics
Chaos theory
Dynamical systems
Social sciences
Economics
Population dynamics
List of named differential equations
Classification
Types
Ordinary
Partial
Differential-algebraic
Integro-differential
Fractional
Linear
Non-linear
By variable type
Dependent and independent variables
Autonomous
Coupled / Decoupled
Exact
Homogeneous / Nonhomogeneous
Features
Order
Operator
Notation
Relation to processes
Difference (discrete analogue)
Stochastic
Stochastic partial
Delay
Solution
Existence and uniqueness
Picard–Lindelöf theorem
Peano existence theorem
Carathéodory's existence theorem
Cauchy–Kowalevski theorem
General topics
Initial conditions
Boundary values
Dirichlet
Neumann
Robin
Cauchy problem
Wronskian
Phase portrait
Lyapunov / Asymptotic / Exponential stability
Rate of convergence
Series / Integral solutions
Numerical integration
Dirac delta function
Solution methods
Inspection
Method of characteristics
Euler
Exponential response formula
Finite difference (Crank–Nicolson)
Finite element
Infinite element
Finite volume
Galerkin
Petrov–Galerkin
Green's function
Integrating factor
Integral transforms
Perturbation theory
Runge–Kutta
Separation of variables
Undetermined coefficients
Variation of parameters
People
List
Isaac Newton
Gottfried Leibniz
Jacob Bernoulli
Leonhard Euler
Joseph-Louis Lagrange
Józef Maria Hoene-Wroński
Joseph Fourier
Augustin-Louis Cauchy
George Green
Carl David Tolmé Runge
Martin Kutta
Rudolf Lipschitz
Ernst Lindelöf
Émile Picard
Phyllis Nicolson
John Crank
v
t
e
In control theory, a continuous linear time-invariant system (LTI) is exponentially stable if and only if the system has eigenvalues (i.e., the poles of input-to-output systems) with strictly negative real parts (i.e., in the left half of the complex plane).[1] A discrete-time input-to-output LTI system is exponentially stable if and only if the poles of its transfer function lie strictly within the unit circle centered on the origin of the complex plane. Systems that are not LTI are exponentially stable if their convergence is bounded by exponential decay.
Exponential stability is a form of asymptotic stability, valid for more general dynamical systems.
^David N. Cheban (2004), Global Attractors Of Non-autonomous Dissipative Dynamical Systems. p. 47
and 26 Related for: Exponential stability information
not LTI are exponentially stable if their convergence is bounded by exponential decay. Exponentialstability is a form of asymptotic stability, valid for...
of exponentialstability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can...
is called linearly stable. Other names for linear stability include exponentialstability or stability in terms of first approximation. If there exists...
Look up backoff in Wiktionary, the free dictionary. Exponential backoff is an algorithm that uses feedback to multiplicatively decrease the rate of some...
and the nearby points converge to it at an exponential rate, cf Lyapunov stability and exponentialstability. If none of the eigenvalues are purely imaginary...
sequence Exponential smoothing ExponentialstabilityExponential sum Exponential time Sub-exponential time Exponential tree Exponential type Exponentially equivalent...
costly. Other modifications of the Euler method that help with stability yield the exponential Euler method or the semi-implicit Euler method. More complicated...
processes are those that possess null recurrent classes. Lyapunov stabilityExponentialstability Gene F. Franklin; J. David Powell; Abbas Emami-Naeini (2006)...
visible from the phase line. The simplest non-trivial examples are the exponential growth model/decay (one unstable/stable equilibrium) and the logistic...
Cauchy problem Wronskian Phase portrait Lyapunov / Asymptotic / Exponentialstability Rate of convergence Series / Integral solutions Numerical integration...
Cauchy problem Wronskian Phase portrait Lyapunov / Asymptotic / Exponentialstability Rate of convergence Series / Integral solutions Numerical integration...
In mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio...
ordinary differential equations (ODEs). The phase portrait can indicate the stability of the system. The phase portrait behavior of a system of ODEs can be...
\cos(px-p\alpha )\ .} Later, Augustin Cauchy expressed the theorem using exponentials: f ( x ) = 1 2 π ∫ − ∞ ∞ e i p x ( ∫ − ∞ ∞ e − i p α f ( α ) d α )...
Cauchy problem Wronskian Phase portrait Lyapunov / Asymptotic / Exponentialstability Rate of convergence Series / Integral solutions Numerical integration...
Cauchy problem Wronskian Phase portrait Lyapunov / Asymptotic / Exponentialstability Rate of convergence Series / Integral solutions Numerical integration...
Cauchy problem Wronskian Phase portrait Lyapunov / Asymptotic / Exponentialstability Rate of convergence Series / Integral solutions Numerical integration...
dx}\,dx\right)+C} where C {\displaystyle C} is a constant. Moving the exponential to the right-hand side, the general solution to Ordinary Differential...
Cauchy problem Wronskian Phase portrait Lyapunov / Asymptotic / Exponentialstability Rate of convergence Series / Integral solutions Numerical integration...
differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence and smoothness of solutions...
Cauchy problem Wronskian Phase portrait Lyapunov / Asymptotic / Exponentialstability Rate of convergence Series / Integral solutions Numerical integration...
has no non-physical energy growth. This guarantees stability if an integration scheme with a stability region that includes parts of the imaginary axis...
Cauchy problem Wronskian Phase portrait Lyapunov / Asymptotic / Exponentialstability Rate of convergence Series / Integral solutions Numerical integration...
Cauchy problem Wronskian Phase portrait Lyapunov / Asymptotic / Exponentialstability Rate of convergence Series / Integral solutions Numerical integration...
integration, and contains many usual functions and special functions such as exponential function, logarithm, sine, cosine, inverse trigonometric functions, error...
Global Information