# The Azimuth Project Blog - categories in control (part 4) (changes)

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## Categories in Control (Part 4): Behavioural controllability

Now we view a system as a set of trajectories. But control theory is about controlling systems. What does it mean to have control of a system? One definition, proposed by the great Dutch control theorist Jan Willems, is that a system is controllable if

1. given some current trajectory $c$ and some present time $t$
2. you pick any other trajectory $d$
3. you can find a trajectory $w$ that is the same as $c$ until time $t$, and after some finite time later $t'$, is the same as $d$.

This says that

But what are the axioms for turning cospans into spans? Bonchi, Sobocinski, and Zanasi proved that they are