Delay differential equations with distributed time delays

Consider delay differential equations with distributed time delays in the form

\[ \dot x(t) = f(x(t), z(t)), \quad z(t) = \int_{-\infty}^t \alpha(t - s) x(s) \, \mathrm ds, \quad t \geq t_0, \]

where \(x\) is the state, \(z\) is the memory state, \(f\) is the right-hand side function, and \(\alpha\) is a kernel or memory function that specifies the weight of states in the past (relative to the current time, \(t\)).

Essentially, this system can be approximated by approximating the convolution in the definition of the memory state. Specifically, we can approximate the kernel, \(\alpha\), with a Erlang mixture approximation in order to approximate the memory state, \(z\), by the weighted sum

\[ \hat z(t) = \sum_{m=0}^M c_m z_m(t), \]

where the auxiliary memory states, \(z_m\) for \(m = 0, \ldots, M\), are obtained as the solution to the initial value problem

\[ \begin{align*} z_m(t_0) &= \int_{-\infty}^{t_0} \ell_m(t_0 - s) x(s) \, \mathrm ds, \\ \dot z_0(t) &= a (x(t) - z_0(t)), & t &\geq t_0, \\ \dot z_m(t) &= a (z_{m-1}(t) - z_m(t)), & t &\geq t_0, & m = 1, \ldots, M. \end{align*} \]

To summarize, delay differential equations with distributed time delays can be approximated by the system

\[ \begin{align*} \dot{\hat x}(t) &= f(\hat x(t), \hat z(t)), & t &\geq t_0, \\ \dot z_0(t) &= a (x(t) - z_0(t)), & t &\geq t_0, \\ \dot z_m(t) &= a (z_{m-1}(t) - z_m(t)), & t &\geq t_0, & m &= 1, \ldots, M, \end{align*} \]

where the coefficients, \(\{c_m\}_{m=0}^M\), and the rate parameter, \(a\), are determined by the Erlang mixture approximation of the kernel, \(\alpha\).