How does the intuitive notion of causality fit in with control systems?
As near as we can tell by experiment, causality is nature's way of doing its thing.
Causality says that if you have a system $y(t) = h\left(x(t), t\right)$, and it is causal, then $y(t_0)$ is dependent only on values of $x(t)$ for $t &lt; t_0$*.
Causality doesn't say you can't combine systems into larger systems (as a rather pertinent example, you can certainly combine systems in a way that makes a control loop).
Causality doesn't imply stability, or instability.  There's a rich set of behaviors that causality allows -- it just doesn't allow the output to follow future inputs.
So in your exponential growth example, if you have a system described by $\dot x = a x + u$, and $a &gt; 0$, the output will tend to grow exponentially (unless $u$ magically or through feedback damps the tendency).  We can say that the growth is caused by $x$ feeding back on itself -- but that causation still happens without violating causality.  Any change in $x$ or $u$ at time $t_0$ will affect the derivative of $x$ at time $t_0$, but (in the absence of infinities) it won't change the immediate value of $x$ -- so, the system is still causal.
* I may be overly strict here -- I'm ruling out $x = u$ as a causal system.  I'm not sure what the popular definition is in control system terms, but I'm lazy so I'm going to let it stand.