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In mathematics, delay differential equations (DDEs) are a type of
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. DDEs are also called time-delay systems, systems with aftereffect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations. They belong to the class of systems with the functional state, i.e. partial differential equations (PDEs) which are infinite dimensional, as opposed to
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 contras ...
(ODEs) having a finite dimensional state vector. Four points may give a possible explanation of the popularity of DDEs: # Aftereffect is an applied problem: it is well known that, together with the increasing expectations of dynamic performances, engineers need their models to behave more like the real process. Many processes include aftereffect phenomena in their inner dynamics. In addition,
actuator An actuator is a component of a machine that is responsible for moving and controlling a mechanism or system, for example by opening a valve. In simple terms, it is a "mover". An actuator requires a control device (controlled by control signal) an ...
s, sensors, and
communication networks A telecommunications network is a group of nodes interconnected by telecommunications links that are used to exchange messages between the nodes. The links may use a variety of technologies based on the methodologies of circuit switching, mess ...
that are now involved in feedback control loops introduce such delays. Finally, besides actual delays, time lags are frequently used to simplify very high order models. Then, the interest for DDEs keeps on growing in all scientific areas and, especially, in control engineering. # Delay systems are still resistant to many ''classical'' controllers: one could think that the simplest approach would consist in replacing them by some finite-dimensional approximations. Unfortunately, ignoring effects which are adequately represented by DDEs is not a general alternative: in the best situation (constant and known delays), it leads to the same degree of complexity in the control design. In worst cases (time-varying delays, for instance), it is potentially disastrous in terms of stability and oscillations. # Voluntary introduction of delays can benefit the control system. # In spite of their complexity, DDEs often appear as simple infinite-dimensional models in the very complex area of partial differential equations (PDEs). A general form of the time-delay differential equation for x(t)\in \R^n is \fracx(t)=f(t,x(t),x_t), where x_t=\ represents the trajectory of the solution in the past. In this equation, f is a functional operator from \R \times \R^n\times C^1(\R, \R^n) to \R^n.


Examples

* Continuous delay \fracx(t)=f\left(t,x(t),\int_^0x(t+\tau)\, d\mu(\tau)\right) * Discrete delay \fracx(t)=f(t,x(t),x(t-\tau_1),\dots,x(t-\tau_m)) for \tau_1 > \dots > \tau_m\geq 0. * Linear with discrete delays \fracx(t)=A_0x(t)+A_1x(t-\tau_1) + \dots + A_mx(t-\tau_m) where A_0,\dotsc,A_m\in \R^. * Pantograph equation \fracx(t) = ax(t) + bx(\lambda t), where ''a'', ''b'' and ''λ'' are constants and 0 < ''λ'' < 1. This equation and some more general forms are named after the pantographs on trains.


Solving DDEs

DDEs are mostly solved in a stepwise fashion with a principle called the method of steps. For instance, consider the DDE with a single delay \fracx(t)=f(x(t),x(t-\tau)) with given initial condition \phi\colon \tau,0to \R^n. Then the solution on the interval ,\tau/math> is given by \psi(t) which is the solution to the inhomogeneous
initial value problem In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or ot ...
\frac\psi(t)=f(\psi(t),\phi(t-\tau)), with \psi(0)=\phi(0). This can be continued for the successive intervals by using the solution to the previous interval as inhomogeneous term. In practice, the initial value problem is often solved numerically.


Example

Suppose f(x(t),x(t-\tau))=ax(t-\tau) and \phi(t)=1. Then the initial value problem can be solved with integration, x(t)=x(0)+ \int_^t \fracx(s) \,ds =1+a\int_^t \phi(s-\tau)\,ds, i.e., x(t)=at+1, where the initial condition is given by x(0)=\phi(0)=1. Similarly, for the interval t\in tau,2\tau/math> we integrate and fit the initial condition, \begin x(t) = x(\tau) + \int_^t \fracx(s) \,ds &= (a\tau+1) + a\int_^t \left(a(s-\tau)+1 \right) ds \\ &= (a\tau+1)+a\int_^ \left(as+1\right) ds, \end i.e., x(t)=(a\tau+1)+a(t-\tau)\left(\frac + 1\right).


Reduction to ODE

In some cases, differential equations can be represented in a format that looks like delay
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s. * Example 1 Consider an equation \fracx(t)=f\left(t,x(t),\int_^0x(t+\tau)e^\,d\tau\right). Introduce y(t)=\int_^0 x(t+\tau)e^\,d\tau to get a system of ODEs \fracx(t)=f(t,x,y),\quad \fracy(t)=x-\lambda y. * Example 2 An equation \fracx(t)=f\left(t,x(t),\int_^0 x(t+\tau)\cos(\alpha\tau+\beta)\,d\tau\right) is equivalent to \fracx(t)=f(t,x,y),\quad \fracy(t)=\cos(\beta)x+\alpha z,\quad \fracz(t)=\sin(\beta) x-\alpha y, where y=\int_^0x(t+\tau)\cos(\alpha\tau+\beta)\, d\tau,\quad z=\int_^0x(t+\tau)\sin(\alpha\tau+\beta)\,d\tau.


The characteristic equation

Similar to
ODE An ode (from grc, ᾠδή, ōdḗ) is a type of lyric poetry. Odes are elaborately structured poems praising or glorifying an event or individual, describing nature intellectually as well as emotionally. A classic ode is structured in three majo ...
s, many properties of linear DDEs can be characterized and analyzed using the characteristic equation. The characteristic equation associated with the linear DDE with discrete delays \fracx(t) = A_0x(t) + A_1x(t-\tau_1) + \dots + A_mx(t-\tau_m) is \det(-\lambda I+A_0+A_1e^+\dotsb+A_me^)=0. The roots ''λ'' of the characteristic equation are called characteristic roots or eigenvalues and the solution set is often referred to as the
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
. Because of the exponential in the characteristic equation, the DDE has, unlike the ODE case, an infinite number of eigenvalues, making a spectral analysis more involved. The spectrum does however have some properties which can be exploited in the analysis. For instance, even though there are an infinite number of eigenvalues, there are only a finite number of eigenvalues to the right of any vertical line in the complex plane. This characteristic equation is a nonlinear eigenproblem and there are many methods to compute the spectrum numerically. In some special situations it is possible to solve the characteristic equation explicitly. Consider, for example, the following DDE: \fracx(t)=-x(t-1). The characteristic equation is -\lambda-e^=0. There are an infinite number of solutions to this equation for complex ''λ''. They are given by \lambda=W_k(-1), where ''W''''k'' is the ''k''th branch of the
Lambert W function In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function , where is any complex number and is the exponential functio ...
.


Applications

* Dynamics of diabetes * Epidemiology * Population dynamics *Classical electrodynamics


See also

*
Functional differential equation A functional differential equation is a differential equation with deviating argument. That is, a functional differential equation is an equation that contains a function and some of its derivatives evaluated at different argument values. Functiona ...
* Halanay Inequality


References


Further reading

* * * * *


External links

* {{scholarpedia, title=Delay-Differential Equations, urlname=Delay-Differential_Equations, curator=Skip Thompson Control theory Differential equations