What is it called if a system is "in rest" until time $t=t_0$? I am a mathematics student, and am quite new to the whole control and system theory (haven't heard any lectures). I currently work with dynamic systems, but do not know the correct words for every attribute, yet.
I would like to describe a system as follows:

A system is in "rest" until time $t=t_0$ for some $t_0\in\mathbb{R}$ if the system response $y(t&lt;t_0)$ and the state $x(t&lt;t_0)$ are equivalent to $0$ for zero input $u(t&lt;t_0) \equiv 0$. 

I am quite sure, that there is a word describing this scenario. 
I know the term causal in terms of systems, but I don't think this is the word I need.

EDIT: To maybe make it more clear what i want:
I want to say "A system is in XXXX, when for zero input there is zero output." (~the system does nothing before any input is applied)
There are two things to consider here. First of all, a non-causal system can react to input that hasn't arrived yet. And, secondly, there could be non-zero initial conditions that would contribute to the output signal without any input signal.

So what you probably mean is a "causal system initially at rest" or "a causal system with zero initial conditions".