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Delay Differential Equations

Prof. Amal Khalaf  Haydar
Department of Mathematics
      In mathematics, delay differential equations (DDEs) are a type of differential equations 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, 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. (2) Delay systems are still resistant to many classical controllers. (3) Delay properties are also surprising since several studies have shown that voluntary introduction of delays can also benefit the control. (4) In spite of their complexity, DDEs however often appear as simple infinite-dimensional models in the complex area of partial differential equations.
A general form of the delay differential equation for x ( t ) ∈ R n {\displaystyle x(t)\in \mathbb {R} ^{n}}    and t  is
 d d t x ( t ) = f ( t , x ( t ) , x t ) , {\displaystyle {\frac {\rm {d}}{{\rm {d}}t}}x(t)=f(t,x(t),x_{t}),}
where  x t = { x ( τ ) : τ ≤ t } {\displaystyle x_{t}=\{x(\tau ):\tau \leq t\}} R × R n × C 1 ( R , R n ) {\displaystyle \mathbb {R} \times \mathbb {R} ^{n}\times C^{1}(\mathbb {R} ,\mathbb {R} ^{n})}   R n . {\displaystyle \mathbb {R} ^{n}.\,} , ,, , and  is a functional operator.
    Similar to ODEs, many properties of linear DDEs can be characterized and analyzed using the characteristic equation. The characteristic equation associated with a linear DDE with discrete delays can be found by using the assumption  . The roots λ of the characteristic equation are called characteristic roots or eigenvalues and the solution set is often referred to as the spectrum[2].
[1] Bellman, Richard; Cooke, Kenneth L. (1963). Differential-difference equations. New York-London: Academic Press. ISBN 978-0-12-084850-8.
[2]  Driver, Rodney D. (1977). Ordinary and Delay Differential Equations. New York: Springer Verlag. ISBN 0-387-90231-7.

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