User:Barbarr/Delay differential equation

In mathematics, delay differential equations (DDEs) are a type of differential equation 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. DDEs are a subclass of functional differential equations.

A general form of the time-delay differential equation for $$\mathbf{x}(t)\in \mathbb{R}^n$$ is
 * $$\frac{\rm d}{{\rm d}t}\mathbf{x}(t)=\mathbf{f}(t,\mathbf{x}(t),\mathbf{x}_t),$$

where $$\mathbf{x}_t=\{\mathbf{x}(\tau):\tau\leq t\}$$ represents the trajectory of the solution in the past. In this equation, $$\mathbf{f}$$ is a functional operator from $$\mathbb{R}\times \mathbb{R}^n\times C^1(\mathbb{R}, \mathbb{R}^n)$$ to $$\mathbb{R}^n.\,$$

Motivation
Four points may give a possible explanation of the popularity of DDEs in various areas of science and engineering:

(Abdallah, Dorato, Benitez-Read, & Byrne, 1993; Richard, Goubet, Tchangani, & Dambrine, 1997, Chap. 11) delayed resonators (Jalili & Olgac, 1998), time-delay controllers and observers (see Section 5.4), nonlinear limit cycle control (Aernouts, Roose, & Sepulchre, 2000), and deadbeat control (Watanabe et al., 1996);
 * 1) Aftereffect is an applied problem: aftereffect phenomena are often present in real-world systems. For example, control system components such as actuators, sensors, and communication networks can introduce delays to feedback control loops. Time lags are also frequently used to simplify very high order models. For these reasons, DDEs are of interest in fields such as control engineering and systems modeling.
 * 2) Delay systems are still resistant to many classical controllers: Ignoring effects which are adequately represented by DDEs and replacing them with finite-dimensional approximations can lead to unexpected effects. In the best cases, where delays are constant and known, it leads to the same degree of complexity in the control design. In the worst cases, where (e.g. where delays vary with time), these approximations can be disastrous in terms of stability and oscillations.
 * 3) Voluntary introduction of delays can benefit control: For example, time-delay controllers
 * 1) In spite of their complexity, DDEs however often appear as simple infinite-dimensional models in the very complex area of partial differential equations (PDEs).

Examples

 * Continuous delay
 * $$\frac{\rm d}{{\rm d}t}x(t)=f\left(t,x(t),\int_{-\infty}^0x(t+\tau)\,{\rm d}\mu(\tau)\right)$$


 * Discrete delay
 * $$\frac{\rm d}{{\rm d}t}x(t)=f(t,x(t),x(t-\tau_1),\dotsc,x(t-\tau_m))$$ for $$\tau_1>\dotsb>\tau_m\geq 0$$.


 * Linear with discrete delays
 * $$\frac{\rm d}{{\rm d}t}x(t)=A_0x(t)+A_1x(t-\tau_1)+\dotsb+A_mx(t-\tau_m)$$
 * where $$A_0,\dotsc,A_m\in \mathbb{R}^{n\times n}$$.


 * Pantograph equation
 * $$\frac{\rm d}{{\rm d}t}x(t) = ax(t) + bx(\lambda t),$$
 * 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


 * $$\frac{\rm d}{{\rm d}t}x(t)=f(x(t),x(t-\tau))$$

with given initial condition $$\phi\colon [-\tau,0]\rightarrow \mathbb{R}^n$$. Then the solution on the interval $$[0,\tau]$$ is given by $$\psi(t)$$ which is the solution to the inhomogeneous initial value problem


 * $$\frac{\rm d}{{\rm d}t}\psi(t)=f(\psi(t),\phi(t-\tau))$$,

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,


 * $$x(t)=x(0)+ \int_{s=0}^t \frac{\rm d}{{\rm d}t}x(s) \,{\rm d}s =1+a\int_{s=0}^t \phi(s-\tau)\,{\rm d}s,$$

i.e., $$x(t)=at+1$$, where the initial condition is given by $$x(0)=\phi(0)=1$$. Similarly, for the interval $$t\in[\tau,2\tau]$$ we integrate and fit the initial condition,



\begin{align} x(t)=x(\tau) & {} + \int_{s=\tau}^t \frac{\rm d}{{\rm d}t}x(s) \,{\rm d}s = (a\tau+1) \\ & {} +a\int_{s=\tau}^t a(s-\tau)+1 \,{\rm d}s = (a\tau+1)+a\int_{s=0}^{t-\tau} as+1 \,{\rm d}s, \end{align} $$

i.e., $$x(t)=(a\tau+1)+a(t-\tau)\left(\frac{a(t-\tau)}2 + 1\right).$$

Reduction to ODE
In some cases, differential equation can be represented in a format that looks like a delay differential equations.
 * Example 1 Consider an equation

\frac{\rm d}{{\rm d}t}x(t)=f\left(t,x(t),\int_{-\infty}^0x(t+\tau)e^{\lambda\tau}\,{\rm d}\tau\right). $$


 * Introduce $$y(t)=\int_{-\infty}^0x(t+\tau)e^{\lambda\tau}\,{\rm d}\tau$$ to get a system of ODEs

\frac{\rm d}{{\rm d}t}x(t)=f(t,x,y),\quad \frac{\rm d}{{\rm d}t}y(t)=x-\lambda y. $$


 * Example 2 An equation

\frac{\rm d}{{\rm d}t}x(t)=f\left(t,x(t),\int_{-\infty}^0x(t+\tau)\cos(\alpha\tau+\beta)\,{\rm d}\tau\right) $$


 * is equivalent to

\frac{\rm d}{{\rm d}t}x(t)=f(t,x,y),\quad \frac{\rm d}{{\rm d}t}y(t)=\cos(\beta)x+\alpha z,\quad \frac{\rm d}{{\rm d}t}z(t)=\sin(\beta) x-\alpha y, $$


 * where

y=\int_{-\infty}^0x(t+\tau)\cos(\alpha\tau+\beta)\,{\rm d}\tau,\quad z=\int_{-\infty}^0x(t+\tau)\sin(\alpha\tau+\beta)\,{\rm d}\tau. $$

The characteristic equation
Similar to ODEs, 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
 * $$\frac{\rm d}{{\rm d}t}x(t)=A_0x(t)+A_1x(t-\tau_1)+\dotsb+A_mx(t-\tau_m)$$

is
 * $${\rm det}(-\lambda I+A_0+A_1e^{-\tau_1\lambda}+\dotsb+A_me^{-\tau_m\lambda})=0$$.

The roots λ of the characteristic equation are called characteristic roots or eigenvalues and the solution set is often referred to as the spectrum. 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:
 * $$\frac{\rm d}{{\rm d}t}x(t)=-x(t-1).$$

The characteristic equation is
 * $$-\lambda-e^{-\lambda}=0.\,$$

There are an infinite number of solutions to this equation for complex λ. They are given by
 * $$\lambda=W_k(-1)$$,

where Wk is the kth branch of the Lambert W function.

Software
In MATLAB, the function dde23 can be used to numerically solve delay differential equations.