= History function of a delay differential equation
{title2=$y(t)=\phi(t),\quad -\tau\leq t\leq0$}
= History functions of a delay differential equation
{synonym}
A <history function of a delay differential equation> supplies the past interval needed to start a <retarded delay differential equation>. A single <initial condition> $y(0)=\phi(0)$ generally leaves earlier delayed values unspecified.<Delay-differential equations by Skip Thompson>[Skip Thompson, Delay-differential equations, Initial History Function: prescribing the solution before the starting time]{ref}
For a <continuous function> history, the natural state at time $t$ is the entire segment
$$
y_t(\theta)=y(t+\theta),\qquad -\tau\leq\theta\leq0.
$$
It belongs to the <function space> $C([-\tau,0],\mathbb R)$, an <infinite-dimensional vector space>: the <functions> $1,\theta,\theta^2,\ldots$ 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 $y'(t)=y(t-1)$ with the two <history functions of a delay differential equation>
$$
\phi_A(s)=1,\qquad \phi_B(s)=1+s,\qquad -1\leq s\leq0.
$$
Both give $y(0)=1$, yet on the first step the respective <derivatives> are $1$ and $t$. Direct <integration> gives
$$
y_A(t)=1+t,\qquad y_B(t)=1+\frac{t^2}{2},\qquad 0\leq t\leq1.
$$
In particular $y_A(1)=2$ while $y_B(1)=3/2$. On the next step,
$$
y_A(t)=1+t+\frac{(t-1)^2}{2},\qquad
y_B(t)=\frac32+(t-1)+\frac{(t-1)^3}{6},\qquad 1\leq t\leq2.
$$
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 $y'(t)=a y(t-1)$ with $a\ne0$, observing the entire first-step <derivative> recovers every history value through $\phi(s)=y'(s+1)/a$ for $-1\leq s\leq0$, using the right <derivative> at $0$. The future can depend on the whole <function>, rather than a finite list of its samples.
\Image[/delay-equation-history-comparison.png]
{title=Equal initial values and different histories give different futures}
{description=Two exact <solutions of a differential equation> for $y'=y(t-1)$. Dashed curves are the prescribed <history functions of a delay differential equation>; solid curves are their exact continuations. Both pass through $(0,1)$. The endpoint values $7/2$ and $8/3$ come from the displayed <polynomials>, with no numerical time stepping.}
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