I am trying to figure out how to formalize a dynamical system whose state vector can change dimensions from one step to the next. For example, I have a process (a discrete-time dynamical system, if you could call it) that at time step $t=k$ has a state vector $x(k)$ that is an $n$-dimensional vector and at time $t=k+1$, the state vector $x(k+1)$ can become an $(n+1)$-dimensional vector depending on the input. In the context of dynamical systems, this is an unusual construct that I am not familiar with. Has anyone ran into this? Any clues as to how one might capture/describe the state space for a process like this?
Addendum:
An alternate form would be if the variable dimensional state, $x(k)$, is re-defined to represent the output $y(k)$ of another system, one whose SS is a fixed dimensional vector space:
$x(k+1)=F(k,x(k),u(k))$ , $x(0)=x_0 \in \mathbb{R^n}$
$y(k) = C(k,x(k),u(k))$ , $y(k) \in \mathbb{R}^{d_k}$
In the context of dynamical systems, it seems like this is a more natural construct to capture variability of the dimension: via the output space rather than the internal SS. This removes the challenges introduced with time incremental change in SS dimension, i.e. $x(k+1) \in \mathbb{R^{n}}$ has the same dimension as $x(k) \in \mathbb{R^{n}}$, but now the output is free to change dimension via an output transition mapping, $C$, in $y(k)=C(k,x(k))$ or more generally $y(k)=C(k,x(k),u(k))$ where $u$ is the input to the system. But now, of course the devil is in figuring out the mapping $C$ that gets us to $x$. I am hoping there are more examples/hits on this form of the problem?
Best Answer
One can consider
$$\cdots\xrightarrow{f_{n-1}}X_n\xrightarrow{f_n}X_{n+1}\xrightarrow{f_{n+1}}\cdots, $$
where $X_n$ is a $d_n$ dimensional space for each $n$. This would allow both increasing dimensions and decreasing dimensions. (If only increasing dimensions ought to be considered, one can take $X_n$'s such that $d_n\leq d_{n+1}$.)
Formally, one can consider the coproduct $X=\bigsqcup_n X_n$ as fibering above the integers $\mathbb{Z}$ and take the family $\{f_n\}_n$ to be generating a cocycle over the right translation action of $\mathbb{Z}$ on itself, that is, we have the dynamical system
$$F:\mathbb{Z}\times X\to X,\, (m, x_n\in X_n)\mapsto f_{n+m-1}\circ f_{n+m-2}\circ\cdots\circ f_{n+1}\circ f_n(x_n)\in X_{n+m}.$$
Similar constructions are used to "autonomize" time-dependent systems (e.g. in random dynamics or nonautonomous ODEs).
Note that this is not the only way to accomplish formalizing the idea of time-dependent dimensions. For instance it might be easier to work with an infinite dimensional space from the get-go (a finite dimensional vector happens to have all but finitely many entries zero in this framework).