  We consider a $n \times n$ nonlinear reaction-diffusion system posed on a
smooth bounded domain $\Omega$ of $\mathbb{R}^N$. This system models reversible
chemical reactions. We act on the system through $m$ controls ($1 \leq m < n$),
localized in some arbitrary nonempty open subset $\omega$ of the domain
$\Omega$. We prove the local exact controllability to nonnegative (constant)
stationary states in any time $T >0$. A specificity of this control system is
the existence of some invariant quantities in the nonlinear dynamics that
prevents controllability from happening in the whole space
$L^\infty(\Omega)^n$. The proof relies on several ingredients. First, an
adequate affine change of variables transforms the system into a cascade system
with second order coupling terms. Secondly, we establish a new
null-controllability result for the linearized system thanks to a spectral
inequality for finite sums of eigenfunctions of the Neumann Laplacian operator,
due to David Jerison, Gilles Lebeau and Luc Robbiano and precise observability
inequalities for a family of finite dimensional systems. Thirdly, the source
term method, introduced by Yuning Liu, Tak\'eo Takahashi and Marius Tucsnak, is
revisited in a $L^{\infty}$-context. Finally, an appropriate inverse mapping
theorem enables to go back to the nonlinear reaction-diffusion system.
