Simultaneous null controllability of a semilinear system of parabolic equations

 ∂y1 ∂t − ∆y1 + f1(y1, y2) = kχω in Q, ∂y2 ∂t − ∆y2 + f2(y1, y2) = kχω in Q, y1 = y2 = 0 on Σ, y1(0) = y 0 1 , y2(0) = y 0 2 in Ω, where fi i = 1, 2, are functions of class C 1 on R, y i ∈ L (Ω) i = 1, 2, k ∈ L(G) represents the control function and χω is the characteristic function of ω, the set where the control is supported. The functions fi i = 1, 2 are assumed to be globally Lipschitz all along the paper, i.e. there exist Ji > 0, i = 1, 2 such that (1.2) |fi(x, y) − fi(z, u)| 6 Ji(‖x− z‖L2(Ω) + ‖y − u‖L2(Ω)), ∀x, y, z, u ∈ L(Ω). Such a system can be met in the field of mathematical biology; we refer to [1]. In this paper, we focus on a simultaneous null controllability problem with constrained state. Let (ej)j=1,...,m be a family of vectors of L(Q). Suppose that: (1.3) the vectors (ejχω)j=1,...,m are linearly independent.