History function of a delay differential equation ($y(t)=\varphi (t),-\tau \le t\le 0$)

A history function of a delay differential equation supplies the past interval needed to start a retarded delay differential equation. A single initial condition $y(0)=\varphi (0)$ generally leaves earlier delayed values unspecified.^[ref]

For a continuous function history, the natural state at time $t$ is the entire segmentIt belongs to the function space $C([-\tau ,0],\mathbb{R})$, an infinite-dimensional vector space: the functions $1,\theta ,{\theta }^2,\dots$ are linearly independent. Shifting $y_t$ forward retains the part of the past that the equation can still read. Thus even one scalar unknown can require a state in an infinite-dimensional vector space.

The insufficiency of one value is visible without any approximation. Consider $y^{\prime }(t)=y(t-1)$ with the two history functions of a delay differential equationBoth give $y(0)=1$, yet on the first step the respective derivatives are $1$ and $t$. Direct integration givesIn particular $y_A(1)=2$ while $y_B(1)=3/2$. On the next step,At $t=2$ they are $7/2$ and $8/3$, a gap of $5/6$. These different futures reflect different prescribed histories; each fully specified problem has a unique solution of a differential equation. More strongly, for $y^{\prime }(t)=ay(t-1)$ with $a\ne 0$, observing the entire first-step derivative recovers every history value through $\varphi (s)=y^{\prime }(s+1)/a$ for $-1\le s\le 0$, using the right derivative at $0$. The future can depend on the whole function, rather than a finite list of its samples.