I am interested in functions of time I call "trajectories" $h$, each in an arbitrary codomain $D$. $D$ is a set.
I wish to express that these trajectories are determined by an initial condition $h_0$ and an evolution/determining function $f$. (almost-)Like a dynamical system. $f$ can take any required form/signature, even complicated. And $f$ is supposed to be given.
I also wish to express that the universe hosting these trajectories, represented by $f$, is gifted with a memory so any value taken by $h$ depends on all past values taken by $h$, and not on future values.
If time were discrete, I would express it this way:
\begin{equation} \newcommand{\funmap}[4]{{% \left\{% \begin{array}{ccl} #1 & \to & \ #2 \\ #3 & \mapsto & \ #4 \end{array} \right. }} h : \funmap{\mathbb{N}}{D}{n}{h(n)} \end{equation} then \begin{align} h(0) &= h_0\\ \forall n \in \mathbb{N}^{*},\ h(n) &= f\left(h_{\big|[0,n[}\right) \text{ or } f\left(h_0, ..., h_{n-1}\right) \text{ ..} \end{align} .. depending on how we define $f$. Anyway, I repeat I'm mostly interested in expriming that:
1. given $f$ and $h_0$, $h$ is (constructively) completely determined (exists + is unique + we can compute it).
2. any value of $h$ depends on all its past values.
In continuous time however, I spontaneously try to express 2. this way:
\begin{equation} h : \funmap{\mathbb{R^+}}{D}{t}{h(t)} \end{equation}
then \begin{align} h(0) &= h_0\\ \forall t \in \mathbb{R}^{+*},\ h(t) &= f\left(h_{\big|[0,t[}\right) \end{align}
.. but I'm not sure that it makes 1. hold. Is 1. true then?
If not, how to express both 1. and 2. in continuous time?
I am aware that this sounds like a recursive definition over $\mathbb{R}^+$, which I think is impossible.. unless we accept the axiom of choice and make it a transfinite recursion by forcing $\mathbb{R}^+$ to be well-ordered? Which is not constructive..
Is there a better way to write this?
Tracks I could think of:
discretizing $\mathbb{R}^+$, defining $h$ recursively on small bits.. then limit-reducing the size of the bits to zero. But I am afraid that everything would collapse at the limit resulting in $h$ being undefined.. would it be the case?
generalizing the concept of "derivative" and "local variation" to any arbitrary set $D$. Then expressing 1. and 2. like a generalized version of an ODE. I like this option best, and I am aware that it necessitate some kind of continuity in $h$ that cannot be defined for any codomain $D$. But I have a couple of hypotheses we could make us of for this track:
Here are two interesting hypothese on $h$, that may help us:
- if $D$ topological, then $h$ is piecewise-continuous, and every piece interval has positive lenght.
- if not, then $h$ is piecewise-constant, and every piece interval has positive length as well.
Am I looking for something impossible to express?
Can I exploit these hypotheses in a useful way to express 1. and 2.?
This is what I came up with. I'll accept my own answer once someone makes me confident it's valid.
First, property "2.: $h$ depends on its past values" is a dumb property since it is always verified. I.e. for any given $h$, one may build the function $f$ so that $h(t) = f\left(h_{\big|[0,\ t[}\right)$.
However, we may approximate $h$ by a sequence $u$ for an arbitrarily accurate precision $\varepsilon$, then state than "$f$ determines $h$ from its past" if and only if the series it determines is the approximation of $h$ for all $\varepsilon$.
For any precison $\varepsilon\in\mathbb{R}^{+*}$, let ${}^\varepsilon h$ be the sequence approximating $h$ in the sense that:
\begin{equation} \forall n \in \mathbb{N},\quad {}^\varepsilon h_n = h(n\varepsilon) \end{equation}
Now, let's consider $F:\left\{\begin{array}{lcl} \mathbb{R}^{+*} &\to& \mathcal{F}\!\left(\bigcup_{i\in\mathbb{N}}D^i\to D\right)\\ \varepsilon &\mapsto& F(\varepsilon) = f_\varepsilon \end{array}\right.$
Then, once given $h_0 = h(0) \in D$, any $F(\varepsilon)$ defines a sequence $f_\varepsilon\times h_0 = (u_n)_{n\in\mathbb{N}}$ by recursion with:
\begin{align} u_0 &= h_0\\ \forall n \in \mathbb{N}^*,\ u_n &= f_\varepsilon(u_0,\ \dots,\ u_{n-1})\\ \end{align}
So I can finally state than
\begin{equation} «\ F \text{ determines } h \text{ from its past } » \iff \forall \varepsilon \in \mathbb{R}^{+*},\ f_\varepsilon\times h_0 = {}^\varepsilon h \end{equation}
This way, I do have:
1. Given $f_\varepsilon$ and $h_0$, ${}^\varepsilon h$ is entirely defined. So given "$F$ determining $h$ from its past", I can approximate $h$ with arbitrary precision.
2. Any "present" approximation ${}^\varepsilon h_n$ is constructed from its "past" ${}^\varepsilon h_n = f_\varepsilon({}^\varepsilon h_0,\ \dots,\ {}^\varepsilon h_{n-1})$.