Delay differential equation explained

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. 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 (ODEs) having a finite dimensional state vector. Four points may give a possible explanation of the popularity of DDEs:^[1]

1. 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, actuators, sensors, and communication networks 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.
2. 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.
3. Voluntary introduction of delays can benefit the control system.^[2]
4. 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$$\backslash fracx(t)=f(t,x(t),x\_t),$$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 \R x \R ^{n x} C ^{1(\R,} \R ^{n)} to \R ^{n.}

Examples

• Continuous delay $$\backslash fracx(t)=f\backslash left(t,x(t),\backslash int\_^0x(t+\backslash tau)\backslash ,\; d\backslash mu(\backslash tau)\backslash right)$$
• Discrete delay $$\backslash fracx(t)=f(t,x(t),x(t-\backslash tau\_1),\backslash dots,x(t-\backslash tau\_m))$$ for

\tau _{1} > ... > \tau _{m\geq} 0 .

• Linear with discrete delays $$\backslash fracx(t)=A\_0x(t)+A\_1x(t-\backslash tau\_1)\; +\; \backslash dots\; +\; A\_mx(t-\backslash tau\_m)$$ where

A _{0,...c,A} _{m\in} \R ^{n x} .

• Pantograph equation $$\backslash fracx(t)\; =\; ax(t)\; +\; bx(\backslash lambda\; t),$$ 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.^{[3] [4]}

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$$\backslash fracx(t)=f(x(t),x(t-\backslash tau))$$

with given initial condition

\phi\colon [ - \tau, 0 ]\to \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$$\backslash frac\backslash psi(t)=f(\backslash psi(t),\backslash phi(t-\backslash tau)),$$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,

$$x(t)=x(0)+\; \backslash int\_^t\; \backslash fracx(s)\; \backslash ,ds\; =1+a\backslash int\_^t\; \backslash phi(s-\backslash tau)\backslash ,ds,$$

i.e.,

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

i.e., $x(t)=(a\backslash tau+1)+a(t-\backslash tau)\backslash left(\backslash frac\; +\; 1\backslash right).$

Reduction to ODE

In some cases, differential equations can be represented in a format that looks like delay differential equations.

• 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} _{-inf ty} x(t + \tau)e ^{λ\tau} d\tau to get a system of ODEs $$\backslash fracx(t)=f(t,x,y),\backslash quad\; \backslash fracy(t)=x-\backslash lambda\; y.$$

• 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 $$\backslash fracx(t)=f(t,x,y),\backslash quad\; \backslash fracy(t)=\backslash cos(\backslash beta)x+\backslash alpha\; z,\backslash quad\; \backslash fracz(t)=\backslash sin(\backslash beta)\; x-\backslash alpha\; y,$$ where $$y=\backslash int\_^0x(t+\backslash tau)\backslash cos(\backslash alpha\backslash tau+\backslash beta)\backslash ,\; d\backslash tau,\backslash quad\; z=\backslash int\_^0x(t+\backslash tau)\backslash sin(\backslash alpha\backslash tau+\backslash beta)\backslash ,d\backslash tau.$$

The characteristic equation

Similar to ODEs, many properties of linear DDEs can be characterized and analyzed using the characteristic equation.^[5] The characteristic equation associated with the linear DDE with discrete delays$$\backslash fracx(t)\; =\; A\_0x(t)\; +\; A\_1x(t-\backslash tau\_1)\; +\; \backslash dots\; +\; A\_mx(t-\backslash tau\_m)$$is$$\backslash det(-\backslash lambda\; I+A\_0+A\_1e^+\backslash dotsb+A\_me^)=0.$$

The roots λ of the characteristic equation are called characteristic roots or eigenvalues and the solution set is often referred to as the spectrum. 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 in any vertical strip of the complex plane.^[6]

This characteristic equation is a nonlinear eigenproblem and there are many methods to compute the spectrum numerically.^{[7] [8]} In some special situations it is possible to solve the characteristic equation explicitly. Consider, for example, the following DDE:$$\backslash fracx(t)=-x(t-1).$$The characteristic equation is$$-\backslash lambda-e^=0.$$There are an infinite number of solutions to this equation for complex λ. They are given by$$\backslash lambda=W\_k(-1),$$where W_k is the kth branch of the Lambert W function, so:$$x(t)=x(0)\backslash ,\; e^.$$

Another example

The following DDE:^[9] $$\backslash fracu(t)=2u(2t+1)-2u(2t-1).$$

Have as solution in

R the function:^[10] with F(t) the Fabius function.

Applications

• Dynamics of diabetes^[11]
• Epidemiology^{[12] [13]}
• Population dynamics^{[14] [15]}
• Classical electrodynamics^[16]

See also

• Functional differential equation
• Halanay Inequality

Further reading

• Book: Bellen. Alfredo. Numerical Methods for Delay Differential Equations. Zennaro. Marino. Oxford University Press. 2003. 978-0198506546. Numerical Mathematics and Scientific Computation. Oxford, UK.
• Book: Bellman. Richard. Differential-Difference Equations. Cooke. Kenneth L.. Academic Press. 1963. 978-0120848508. Mathematics in Science and Engineering. New York, NY.
• Book: Briat. Corentin. Linear Parameter-Varying and Time-Delay Systems: Analysis, Observation, Filtering & Control. Springer-Verlag. 2015. 978-3662440490. Advances in Delays and Dynamics. Heidelberg, DE.
• Book: Driver. Rodney D.. Ordinary and Delay Differential Equations. Springer-Verlag. 1977. 978-0387902319. Applied Mathematical Sciences. 20 . 10.1007/978-1-4684-9467-9 . New York, NY.
• Book: Erneux. Thomas. Applied Delay Differential Equations. Springer Science+Business Media. 2009. 978-0387743714. Surveys and Tutorials in the Applied Mathematical Sciences. 3 . 10.1007/978-0-387-74372-1 . New York, NY.

Notes and References

1. Richard . Jean-Pierre . Time Delay Systems: An overview of some recent advances and open problems. 2003 . Automatica . 39 . 10 . 1667–1694 . 10.1016/S0005-1098(03)00167-5 . refRichard2003.
2. Book: Lavaei. Javad. Sojoudi. Somayeh. Murray. Richard M.. Proceedings of the 2010 American Control Conference . Simple delay-based implementation of continuous-time controllers . 2010. https://ieeexplore.ieee.org/document/5530439. 5781–5788. 10.1109/ACC.2010.5530439. 978-1-4244-7427-1 . 1200900 .
3. Griebel. Thomas. 2017-01-01. The pantograph equation in quantum calculus. Masters Theses.
4. Ockendon. John Richard. Tayler. A. B.. Temple. George Frederick James. 1971-05-04. The dynamics of a current collection system for an electric locomotive. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 322. 1551. 447–468. 10.1098/rspa.1971.0078. 1971RSPSA.322..447O. 110981464.
5. Book: Michiels. Wim. Niculescu. Silviu-Iulian. Stability and Stabilization of Time-Delay Systems. Society for Industrial and Applied Mathematics. 2007. 978-0-89871-632-0. Advances in Design and Control. 3–32. 10.1137/1.9780898718645.
6. Book: Michiels. Wim. Niculescu. Silviu-Iulian. Stability and Stabilization of Time-Delay Systems. Society for Industrial and Applied Mathematics. 2007. 978-0-89871-632-0. Advances in Design and Control. 9. 10.1137/1.9780898718645.
7. Book: Michiels. Wim. Stability and Stabilization of Time-Delay Systems. Niculescu. Silviu-Iulian. Society for Industrial and Applied Mathematics . 2007. 978-0-89871-632-0. Advances in Design and Control. 33–56. 10.1137/1.9780898718645.
8. Appeltans . Pieter . Michiels . Wim . 2023-04-29 . Analysis and controller-design of time-delay systems using TDS-CONTROL. A tutorial and manual . math.OC . 2305.00341 .
9. 1702.06487 . Juan Arias de Reyna . Arithmetic of the Fabius function . 2017 . math.NT .
10. Web site: A288163 - Oeis .
11. Makroglou. Athena. Li. Jiaxu. Kuang. Yang. 2006-03-01. Mathematical models and software tools for the glucose-insulin regulatory system and diabetes: an overview. Applied Numerical Mathematics. Selected Papers, The Third International Conference on the Numerical Solutions of Volterra and Delay Equations. en. 56. 3. 559–573. 10.1016/j.apnum.2005.04.023. 0168-9274.
12. Salpeter. Edwin E.. Salpeter. Shelley R.. 1998-02-15. Mathematical Model for the Epidemiology of Tuberculosis, with Estimates of the Reproductive Number and Infection-Delay Function. American Journal of Epidemiology. en. 147. 4. 398–406. 10.1093/oxfordjournals.aje.a009463. 9508108 . 0002-9262. free.
13. Kajiwara. Tsuyoshi. Sasaki. Toru. Takeuchi. Yasuhiro. 2012-08-01. Construction of Lyapunov functionals for delay differential equations in virology and epidemiology. Nonlinear Analysis: Real World Applications. en. 13. 4. 1802–1826. 10.1016/j.nonrwa.2011.12.011. 1468-1218.
14. Book: Gopalsamy, K.. Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic Publishers. 1992. 978-0792315940. Mathematics and Its Applications. Dordrecht, NL. 10.1007/978-94-015-7920-9 .
15. Book: Kuang, Y.. Delay Differential Equations with Applications in Population Dynamics. Academic Press. 1993. 978-0080960029 . Mathematics in Science and Engineering. San Diego, CA.
16. López. Álvaro G.. 2020-09-01. On an electrodynamic origin of quantum fluctuations. Nonlinear Dynamics. en. 102. 1. 621–634. 10.1007/s11071-020-05928-5. 1573-269X. 2001.07392. 210838940 .