I'm studying control theory from the book "Mathematical Control Theory" by Sontag. I have a question about discrete-time systems, which is defined on page 32.
Definition.
A system $\Sigma=(\mathcal{T},\mathcal{X},\mathcal{U},\phi)$ is called a discrete-time system if $\mathcal{T}=\mathbb{Z}$.
There is a construction of a function in that chapter I would like to understand, which this question will be about.
Let $\Sigma$ be a discrete-time system and define a function on $\mathcal{E}=\mathcal{E}(\phi)=\{(t,x,u)\in\mathbb{Z}\times\mathcal{X}\times\mathcal{U} | (t+1,t,x,u)\in D_\phi\}$ as follows$$P:\mathcal{E}\to\mathcal{X} \\ (t,x,u)\mapsto \phi(t+1,t,x,\omega),$$ where $\omega\in\mathcal{U}^{\{t\}},$ with $\omega(t)=u$.
According to the book, it is possible to recover the system description using the knowledge of $P$; Their example is that if $\omega\in \mathcal{U}^{[t,t+2)}$, then $$\phi(t+2,t,x,\omega)=P(t+1,P(t,x,\omega(t)),\omega(t+1)).$$
I would like to understand why that equality holds and how to generalize that equality to a total description of $\phi$.
Let us start from the right-hand side, and try to do some manipulations in order to arrive at the left-hand side. Since $P(t+1,x,u)=\phi(t+2,t+1,x,\omega)$, we have that $$P(t+1,P(t,x,\omega(t)),\omega(t+1))=\phi(t+2,t+1,P(t,x,\omega(t)),\omega(t+1)),$$ where $x=P(t,x,\omega(t))$. But we also know that $P(t,x,\omega(t))=\phi(t+1,t,x,\omega)$ and applying that, we have $$\phi(t+2,t+1,P(t,x,\omega(t)),\omega(t+1))=\phi(t+2,t+1,\phi(t+1,t,x,\omega),\omega(t+1)).$$ At this point, I would like to write $$(*)\qquad \phi(t+2,t+1,\phi(t+1,t,x,\omega),\omega(t+1))=\phi(t+2,t,x,\omega),$$ and I am not sure how I should motivate that step. It seems almost "intuitively" clear to me, since, on one hand first we say where $\phi$ should map the state between time $t$ and $t+1$, and then what it should do to the state between time $t+1$ and $t+2$, which is the left-hand side. But on the other hand, we could just say what $\phi$ should to $x$ between time $t$ and $t+2$ all at once, which is the right-hand side.
I'm not sure if the last step is possible thanks to the semigroup axiom of a system:
Semigroup axiom.
If $\sigma,\tau,\mu$ are any elements of $\mathcal{T}$ so that $\sigma<\tau<\mu$ If $\omega_1\in\mathcal{U}^{[\sigma,\tau)}$ and $\omega_2\in\mathcal{U}^{[\tau,\mu)}$ and if $x$ is a state so that $$\phi(\tau,\sigma,x,\omega_1)=x_1\quad \text{ and }\quad \phi(\mu,\tau,x_1,\omega_2)=x_2,$$ then $\omega=\omega_1\omega_2$ is also admissible for $x$ and $\phi(\mu,\sigma, x,\omega)=x_2.$
$\text{ }$
In our case, I guess $\sigma=t$, $\tau=t+1$ and $\mu=t+2$ and $x$ would simply be $x$ and $x_1=\phi(t+1,t,x,\omega)$. Hence, by the semigroup axiom, we would have $$\phi(t+1,t,x,\omega_1)=x_1\quad \text{ and }\quad \phi(t+2,t+1,\phi(t+1,t,x,\omega_1),\omega_2)=x_2,$$ but at the same time, according to the axiom, we have the concatenation $\omega=\omega_1\omega_2$ (which is a function $\omega:[t,t+2)\to\mathcal{U}$) satisfying $$\phi(t+2,t,x,\omega)=x_2,$$ which means that $$\phi(t+2,t,x,\omega)=\phi(t+2,t+1,\phi(t+1,t,x,\omega),\omega(t+1)),$$ which (I suppose) proves the fact.
By generalizing the above argument, I think we can write $$\phi(t+3,t,x,\omega)=P(t+2,P(t+1,P(t,x,\omega(t)),\omega(t+1)),\omega(t+2)).$$
And using induction on the time, I think we can describe $\phi$ by $P$ through any time $[t,t+k)$ by a similar construction as above.
Questions.
One of my original question was how to motivate $(*)$, but I think I accidently did it while formulating this question. But I still wonder, is my motivation correct (I may have done something wrong along the way)?
Is it correct to generalize the construction of $\phi$ by $P$ as I did above? My interpretation of this is that I can view $\phi$ as a dynamical system through $P$ by feeding it into itself until we reach the end-time. Is that a correct interpretation?
Thanks for your help!