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.

A general form of the time-delay differential equation for $x(t)\in \mathbb{R}^n$ is

$\frac{\rm d}{{\rm d}t}x(t)=f(t,x(t),x_t),$

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

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

$\frac{\rm d}{{\rm d}t}x(t)=f(t,x(t),x(t-\tau_1),\ldots,x(t-\tau_m))$ for $\tau_1>\ldots>\tau_m\geq 0$.

• Linear with discrete delay

$\frac{\rm d}{{\rm d}t}x(t)=A_0x(t)+A_1x(t-\tau_1)+\ldots+A_mx(t-\tau_m)$

where $A_0,\ldots,A_m\in \mathbb{R}^{n\times n}$.

• 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 $\phi:[-\tau,0]\rightarrow \mathbb{R}^n$. Then the solution on the interval [0,τ] is given by ψ(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 ψ(0) = ϕ(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 − τ)) = ax(t − τ) and ϕ(t) = 1. Then the initial value problem can be solved with integration,

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

i.e., x(t) = at + 1, where we picked C = 1 to fit the initial condition x(0) = ϕ(0). Similarly, for the interval $t\in[\tau,2\tau]$ we integrate and fit the initial condition to find that x(t) = at² / 2 + t + D where D = (a − 1)τ + 1 − aτ² / 2.

Reduction to ODE

In some cases, delay differential equations are equivalent to a system of ordinary 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^[1]. The characteristic equation associated with the linear DDE with discrete delays

$\frac{\rm d}{{\rm d}t}x(t)=A_0x(t)+A_1x(t-\tau_1)+\ldots+A_mx(t-\tau_m)$

is

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

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 a 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^[2]. 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

λ = W_k( − 1),

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

Notes

1. ^ Michiels, Niculescu, 2007 Chapter 1
2. ^ Michiels, Niculescu, 2007Chapter 2

References

• Bellman, Richard; Cooke, Kenneth L. (1963). Differential-difference equations. New York-London: Academic Press. ISBN 978-0120848508. 
• Driver, Rodney D. (1977). Ordinary and Delay Differential Equations. New York: Springer Verlag. ISBN 0387902317. 
• Michiels, Wim and Niculescu, Silviu-Iulian (2007). Stability and stabilization of time-delay systems. An eigenvalue based approach. ISBN 978-0-898716-32-0. 

External links

• Delay-Differential Equations at Scholarpedia, curated by Skip Thompson.