Doubt on the definition of a dynamical system

Question

I am studying control theory and I am focusing on dynamical system. In the introdution of the notes of my professor, it is defined a dynamical system as a system defined by three elements:

1. time
2. a set of functions defined on the time interval, W
3. the behaviour of the system at the given time

I have clear what are the 1. and 3., but I really can't understand what he means by W. Since we are considering a dynamical system in the context of control thoery, I would guess that these are the inputs and the outputs, but what can they be in a general context?

For example, consider the evoulution of a population. This is clearly a dynamical system, and can be modeled as:

$\dot{x}(t)=cx(t)$

in this case, what is the set of function $W$ as specified in 2.?

Answer 1

Consider a slight modification of your system:

$\frac{d}{dt} x(t)=u(t)x(t)$.

Now, I think you would be comfortable with $W=\{u\}$, right? Now consider that we additionally require $u$ to be continuous. Does this change $W$? You will now probably ask: "why should it?"

And now comes the trick: by the same logic, you will have to admit that requiring $u$ to be constant should not change $W$. Now set $u(t)=c$ and you are back at your original example.