  We consider a $4\times4$ nonlinear reaction-diffusion system posed on a
smooth domain $\Omega$ of $\mathbb{R}^N$ ($N \geq 1$) with controls localized
in some arbitrary nonempty open subset $\omega$ of the domain $\Omega$. This
system is a model for the evolution of concentrations in reversible chemical
reactions. We prove the local exact controllability to stationary constant
solutions of the underlying reaction-diffusion system for every $N \geq 1$ in
any time $T >0$. A specificity of this control system is the existence of some
invariant quantities in the nonlinear dynamics. The proof is based on a
linearization which uses return method and an adequate change of variables that
creates crossed diffusion which will be used as coupling terms of second order.
The controllability properties of the linearized system are deduced from
Carleman estimates. A Kakutani's fixed-point argument enables to go back to the
nonlinear parabolic system. Then, we prove a global controllability result in
large time for $1 \leq N \leq 2$ thanks to our local controllabillity result
together with a known theorem on the asymptotics of the free nonlinear reaction
diffusion system.
