Notation and functions in a linear system of ODEs?

Question

I have a linear system of ODEs \begin{align} \dot x(t)&=A(t)x(t)+B(t)u(t) \tag 1\\ y(t)&=C(t)x(t)+D(t)u(t) \tag 2 \end{align} where \begin{align} x&:\mathbb R\rightarrow\mathbb R^n\\ y&:\mathbb R\rightarrow\mathbb R^m\\ u&:\mathbb R\rightarrow\mathbb R^k\\ \end{align} and the matrix-functions are \begin{align} A&:\mathbb R\rightarrow\mathbb R^{n\times n}\\ B&:\mathbb R\rightarrow\mathbb R^{n\times k}\\ C&:\mathbb R\rightarrow\mathbb R^{m\times n}\\ D&:\mathbb R\rightarrow\mathbb R^{m\times k}\\ \end{align}

Q1: Say I want to write $(1)$ and $(2)$ in a more general form, i.e. should I write

\begin{align} \dot x(t)&=f(x(t),u(t)) \tag 3 \\ y(t)&=g(x(t),u(t)) \tag 4 \end{align} Or

\begin{align} \dot x(t)&=f(A(t),B(t),x(t),u(t)) \tag 5 \\ y(t)&=g(C(t),D(t),x(t),u(t)) \tag 6 \end{align} Q2: From the answer in Q1, what is the correct dimension of $f$ and $g$, i.e. \begin{align} f:\mathbb R^?\rightarrow \mathbb R^?\\ g:\mathbb R^?\rightarrow \mathbb R^? \end{align}

Thanks!

Answer 1

Such a choice is arbitrary. Do you view $A$ and $B$ as time varying parameters, or as inputs like $u$? (Probably the former, since you described (1) and (2) as a "linear system"). It's all about context. In control theory, equations (3) and (4) are the most common for a nonlinear ODE system in explicit state-space form. I.e. if the dependent variables and inputs are real curves like you said, \begin{align} x &: \mathbb{R} \to \mathbb{R}^n \\ y &: \mathbb{R} \to \mathbb{R}^m \\ u &: \mathbb{R} \to \mathbb{R}^k \end{align} then the usual situation is, \begin{align} \dot{x} &= f(x, u) \\ y &= g(x, u) \end{align} where, \begin{align} f &: \mathbb{R}^n \times \mathbb{R}^k \to \mathbb{R}^n \\ g &: \mathbb{R}^n \times \mathbb{R}^k \to \mathbb{R}^m \\ \end{align} while $A(t)$, $B(t)$, $C(t)$, $D(t)$ would be hidden away within the workings of these functions as "time varying parameters" or "exogenous inputs." Finally, though I think it'd be misleading, you could write $\mathbb{R}^n \times \mathbb{R}^k$ as $\mathbb{R}^{n+k}$ to fit exactly the notation you are looking for.

This is not "the most general system" one can write though. It need not be explicit in $\dot{x}$ and $y$, need not necessary have a real / Euclidean state (like for example if $x : \mathbb{R} \to \mathbb{SO}3$), and need not even be a finite state system (i.e. it could be a PDE instead of an ODE). Your equations (1) and (2) are just a very very special case. But also a very important one to study :)

Answer 2

A system is linear if $A(t),B(t),C(t),D(t)$ are linear i.e $dx/dt=2x+3u$ . If these depend on time as the questions looks like , then (5) and (6) are correct.

If you look at (5) and (6) $A,B,C,D,x,y$ are independent .To decide to what dimension it goes to , just recall the definition of vector field . For example , $h(x)=f(u(x),h(x))$ here h goes to $R^2$ . Similarly calculate all possible depencies of your $f.f$ goes from $R$ to $R^{n^2+nk+2}$.($n^2$ total no. of tuples in $A$ $nk$ is total no. of tuples in $B$ and others are $x(t)$ and $u(t)$) .