Mathematical definition of a "system" as used in engineering I was taught many years ago that a system $S$ converts an input signal/function $u(t)$ into an output signal/function $y(t)$: $y(t)=S \big( u(t) \big)$. 
So what would be a mathematical definition of the system $S$? Is $S$ itself a function? If so, what would be the domain and codomain? Is it possible to "plot" or somehow visualize this function??
I also remember something to the effect that an LTI (linear time invariant) system is completely specified by its impulse response $h(t)$. What would be in this case the relationship between $S$ and $h(t)$? They cannot be the same because the domain and codomain of $h(t)$ are both $R$ (in the SISO (Single Input Single Output) case to keep things simple), but I think that that can't be true for $S$.
Many thanks in advance for helping me sorting out this confusion.
 A: I think what you are looking for are Dynamical Systems. Here i give an attempt of joining a mathematical definition as in the linked article with those concepts used in engineering, while trying to convince you this is the concept you are looking.
Definition
A System is mainly an "Evolution Function" $\Phi$ between an "Evolution Parameter Space" set $T$ and a "Phase" or "State Space" set $X$ back into the State Space, under a specific subset of $T \times X$ defined for trajectories  $\{x=\Phi(t,x_0), (t,x)\in S, \forall t \in T\}$:
$$ 
\Phi: S \subseteq T \times X \to X \\
x=\Phi(0,x)\\
x_1=\Phi(t_1,x)\\
x_2=\Phi(t_2,x)=\Phi(t_{21},\Phi(t_1,x))\\
(0,x),(t_1,x),(t_2,x),(t_{21},x_1)\in S
$$
Non Dynamical - Static
A Non Dynamical (i.e. Static) System is a very particular kind of system: one which do not depend on (past or present values of) the state:
$$ \Phi(t,x)=\Phi(t), \forall t\in T, \forall x \in X$$
Time Invariant
Similarly, a Time Invariant System is a system in which displacements in the Evolution Parameter do not affect the trajectory:
$$ \Phi(t,x)=\Phi(t+\Delta t,x), \forall t,\Delta t \in T, \forall x \in X$$
Note that Static Time Varying Systems are Constant Gains solely depending on time and Static Time Invariant Systems are simply constant values. Henceforth, the only relevant systems are the Dynamic Systems, because we are not so much interested in describe perfect gains and constants.  
Linear
Thus, the standard continuation are Linear Systems, which are linear in the State variable, when $M$ is equipped with the addition $+$ and scalar ponderation $\cdot$ operators:
$$ 
\Phi(t,a_1\cdot x_1+a_2\cdot x_2)=a_1\cdot\Phi(t,x_1)+a_2\cdot\Phi(t,x_2), \\
\forall t \in T, \forall x_1,x_2 \in X, a_1,a_2 \in $$
Discrete | Continuous
According the Evolution variable, in a Discrete Time System the set $T$ is a disconnected set, or simply, a subset of $\mathbb N_{\ge 0}$, and in a Continuous Time System, the set $T$ is connected, or simply, a subset of $\mathbb R_{\ge 0}$.
Also, usually, $X$ is a Banach Space in $\mathbb R$ with the Euclidian Norm $|\cdot|$.
Order
Regarding Dynamical Systems, it is more than habitual to define the Space set $X$ by including the values of "n" derivatives of a given variable. In this sense, if $\Phi$ do not depend on other derivatives rather than the variables defined for the set $X$, "n" is the "Order" of the system. This definition, as we know, is rather algebraic rather than practical. One can never be sure a system has "n" derivatives...
$$ X=(x,x',x'',...,x^{(n)})$$
From here, people sometimes agrees on simplify the notation of system by a single function not mapping back into the State set, but as a implicit definition:
$$ \Phi(t,x)=0$$
or making explicit a "derivative"|"difference" of the state variable:
$$ x'=\Phi(t,x) \\
 \Delta x=\Phi(t,x)$$
Autonomous | Exogenous
Some physics works with Autonomous Dynamical Systems, as systems evolving solely based on their state, as we have already defined. 
$$x'=\Phi(t,x)$$
And engineers prefers the Exogenous Dynamical Systems, introducing an additional input variable $u$, which is nothing more than a input state, 
$$x'=\Phi(t,x,u)$$
and even some creatives adding noise "e", outputs "y", perturbations "p", some hidden the states, and so:
$$y=\Phi(t,u)$$
In here, the variables start to show explicitly the dependence on the Evolution variable: $x(t):\{x=\Phi(t,x_0), (t,x)\in S\}$, and similarly for $u(t)$, $y(t)$, relaxing the formality that $(t,[x \ u\  y]) \in S$. 
Also, the concept of Dependency is implied in Control Systems, by expressing the controlled variable $y(t)$ "depends" on the manipulated variable $u(t)$, concept which in industrial application is rather practical ("acting a controller") than theoretical ("causality and correlation").
Signals
Under the Input Output approach, we could think about the set of possible trajectories $x(t):\{x=\Phi(t,x_0),(t,x)\in S\}$ as members of a Hilbert Space in $X$, with $T$ the dimension index variable, and equipped with a norm $|\cdot|$ for measuring the distance between trajectories. 
In the most standard setups, $X$ is $\mathbb R$, $T$ is $\mathbb R_{\ge 0}$ (Continuous) or $\mathbb N_{\ge 0}$ (Discrete), equipped with inner product $|\cdot|$ the integral|sum of the product:
$$
\phi_1\cdot\phi_2=\int_0^{\infty}\phi_1(t)\phi_2(t) dt\\
\phi_1\cdot\phi_2=\sum_{t=0}^{\infty}\phi_1(t)\phi_2(t)
$$
In addition to the Hilbert Space framework, we call the trajectories "signals", as possible outputs from a given system, and in contrast with the inner product, we have the Laplace|Z Transform, their particular cases the Fourier|Discrete Fourier Transforms, the Convolution and Correlation operators, among others.
Transfer Function
Finally, the Transfer Function arises, which is nothing more than the relation between an output $y(t)$ with respect to an input $u(t)$, after taking the Laplace|Z Transform variable $s|z$:
$$ H_{yu}(s)={\mathcal L y(t) \over \mathcal L u(t)}={Y(s)\over U(s)}, (t,[u \ y])\in S$$
And of course, in the Linear Time Invariant (LTI) case, this Transfer Function is independent of both outputs and inputs signals, and equal to the Transform of the Impulse Response of the system. 
$$
H_{yu}(s)=H(s)=\mathcal L h(t)\\
Y(s)=H(s)U(s)\\
y(t)=h(t)*u(t)=\int_{-\infty}^\infty h(t-\tau)u(\tau)d\tau
$$
