Delay Differential Equations II

In the note Delay Differential Equations I we introduced a simple algorithm for solving delay equations of the form

\begin{equation} \label{eq:dde-std} \frac{\mathrm{d}x}{\mathrm{d}t} = f(t, x(t), x(t - \tau)) \end{equation}

This method has its limitations. For instance, it cannot address equations with “distributed delay” such as

\begin{equation} \label{eq:dde-dist} \frac{\mathrm{d}x}{\mathrm{d}t} = f \left(t, x(t), \int_{-\tau}^{0} g(x(t - \sigma)) ~\text{d}\sigma \right) \end{equation}

In this note we give a definition of delay differential equation that includes equations like \ref{eq:dde-std} and \ref{eq:dde-dist} as special cases. We also outline a theoretical framework for the analysis of DDEs.

Abstract definition of a DDE

Let \(C _ {d,\tau}\) denote the space \(C([-\tau, 0] ; \mathbb{R}^d)\) of continuous functions. Fix a continuously Frechet differentiable function \(f \colon C_{d,\tau} \to C_{d,\tau}\) satisfying \(f(0) = 0\). Given some \(x \colon (\alpha, \beta) \to \mathbb{R}^d\), where \((\alpha, \beta)\) is an interval containing \([-\tau, 0]\), we can construct a family of new functions \(x_t \colon [-\tau, 0] \to \mathbb{R}^d\) according to the formula

\begin{equation*} x_t(\sigma) ~:= x(t + \sigma) \end{equation*}

For a given value of \(t\), the function \(x_t\) represents the history of the function \(x\) on the interval \([t-\tau, t]\).

Definition 1. By the term delay differential equation we shall mean an equation of the form

\begin{equation} \label{eq:absdde} \frac{\mathrm{d}x_t}{\mathrm{d}t} = f(x_t) \end{equation}

Note that on the left hand side we are taking a Frechet derivative in the space \(C_{\tau, d}\). As before, we typically impose an initial function condition of the form \(x| _ {[-\tau, 0]} = \varphi\) for some \(\varphi \in C_{\tau, d}\). We shall denote a solution to equation \ref{eq:absdde} with initial function \(\varphi\) by \(x_t^\varphi\). ∎

The important thing distinguishing a DDE from an ODE is that the evolution at time \(t\) depends on the history of the solution on the entire interval \([t-\tau, t]\).

DDE Semigroups

Denote by \(L\) the linearization (Frechet derivative) of the operator \(f\) at the zero function, and let us consider the simplified system

\begin{equation}\label{eq:lin} \frac{\mathrm{d}x}{\mathrm{d}t} = L[x_t], \quad x_0 = \varphi \end{equation}

As usual we can learn a lot about a DDE by studying the flow of solutions to its linearization.

Definition 2. Suppose we are given a linearized DDE as in equation \ref{eq:lin}. For each \(t > 0\), the solution operator \(P[t] \colon C_{\tau, d} \to C_{\tau, d}\) is defined according to the formula

\begin{equation*} P[t](\varphi) = x_t^\varphi \end{equation*}

∎

It is easy to verify the solution operators satisfy the axioms of a one-parameter semigroup. In other words, the solution operators satisfy the following three properties:

  • For all \(t \geq 0\), \(P[t]\) is bounded linear operator on \(C_{\tau, d}\) satisfying
    \begin{equation*} P[s + t](\varphi) = P[s](P[t](\varphi)); \end{equation*}
  • \(P[0](\varphi) = \varphi\);
  • \(\lim_{s \rightarrow t} | P[s](\varphi) - P[t](\varphi) | = 0\).