Distributed delays through a finite-window history integral

Let $\tau >0$, $T>0$, $d\ge 1$, let $K:[0,\tau ]\to \mathbb{R}$ be continuous, and let $y:[-\tau ,T]\to {\mathbb{R}}^d$ be continuous. The distributed-delay history functional is ${\mathcal{I}}_K[y](t):={\int }_0^{\tau }K(\theta )y(t-\theta )d\theta (0\le t\le T)\mathrm{.}$ The integral is the componentwise vector-valued Riemann integral of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral. For each fixed $t\in [0,T]$, $\theta \mapsto t-\theta$ is continuous and takes $[0,\tau ]$ into $[-\tau ,T]$. By A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions, each component $y_i$ is continuous. The map $\theta \mapsto t-\theta$ is polynomial, hence continuous by Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, and A composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs makes each scalar component $\theta \mapsto y_i(t-\theta )$ continuous; Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function then shows that multiplying by $K$ preserves continuity. Since $\tau >0$, each resulting component is bounded and Riemann integrable on $[0,\tau ]$ by A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion. The componentwise definition therefore makes the displayed vector-valued integral well-defined. A distributed-delay equation uses this functional as an input, for example with a continuous $f:U\times {\mathbb{R}}^d\times {\mathbb{R}}^d\to {\mathbb{R}}^d$, where $U\subseteq \mathbb{R}$ is an open interval containing $[0,T]$. For a path whose restriction to $[0,T]$ is continuously differentiable, the equation is $y^{\mathrm{\prime }}(t)=f(t,y(t),{\mathcal{I}}_K[y](t)),0\le t\le T\mathrm{.}$ Derivatives at $0$ and $T$ are taken from within $[0,T]$, as in The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral.

The quadrature rule on the A page is the composite midpoint rule for this specified finite window. Its history interpolation error and its quadrature remainder are separate terms; no order is asserted for a kernel or history without the regularity hypotheses stated in the quadrature lemma.