  We introduce a new setting where a population of agents, each modelled by a
finite-state system, are controlled uniformly: the controller applies the same
action to every agent. The framework is largely inspired by the control of a
biological system, namely a population of yeasts, where the controller may only
change the environment common to all cells. We study a synchronisation problem
for such populations: no matter how individual agents react to the actions of
the controller , the controller aims at driving all agents synchronously to a
target state. The agents are naturally represented by a non-deterministic
finite state automaton (NFA), the same for every agent, and the whole system is
encoded as a 2-player game. The first player (Controller) chooses actions, and
the second player (Agents) resolves non-determinism for each agent. The game
with m agents is called the m-population game. This gives rise to a
parameterized control problem (where control refers to 2 player games), namely
the population control problem: can Controller control the m-population game
for all $m $\in$ N$ whatever Agents does? In this paper, we prove that the
population control problem is decidable, and it is a EXPTIME-complete problem.
As far as we know, this is one of the first results on parameterized control.
Our algorithm, not based on cutoff techniques, produces winning strategies
which are symbolic, that is, they do not need to count precisely how the
population is spread between states. We also show that if there is no winning
strategy, then there is a population size M such that Controller wins the
m-population game if and only if $m $\le$ M$. Surprisingly, M can be doubly
exponential in the number of states of the NFA, with tight upper and lower
bounds.
