Approximate controllability of a reaction-diffusion system

An open-loop control for a system coupling a reaction-diffusion system and an ordinary differential equation is proposed in this study. We use a flatness-like property, indeed, the solution can be expressed in terms of an infinite series depending on a flat output, its derivatives and its integrals. This series is shown to be convergent if the flat output is Gevrey of order 1 < a ≤ 2. Approximate controllability of the system is then proved. We study the approximate controllability of the following reaction-diffusion system      y t = y xx + µy x + Ay + Bz, z t = δz xx + C y + dz, (1) where, y ∈ R p is the vector of diffusing species and z ∈ R the stored one. The parameters A, G ∈ R p×p , B ∈ R p×1 , C ∈ R 1×p , µ, δ, d ∈ R are given. The time dependent vector function u is the control in L 2 (0, T) p. This type of system appears in varied domains such as chemistry , electrophysiology, genetics, combustion...The degenerate case, δ = 0 models, for example, the linearized FitzHugh Nagumo equations in electrophysiology [3,13] where y is the electrical potential and z the chemical concentration. Furthermore, the governing equations for chemical reactions in a tubular reactor model can exactly be written as (1) where y stands for the temperature (p = 1) and z for the solid fuel concentration. In solid combustion, the reactants do not diffuse and thus δ = 0, [11]. Note also that gas electrodes may be modelled by (1) where y is the concentration of diffusive species with p ≤ 4 and z is the concentration of the stored one, [5]. The approximate controllability of the non degenerate case, i.e. δ = 0, has been studied in detail in [4,8] for the linear case and in [12] for a special nonlinear case, see also [2]. They have introduced some Gevrey functions in order to exhibit a ''flat'' output for their equations. The null controllability of the non degenerate case has also been studied with Carleman estimates in [6,7]. A previous study of controllability of the FitzHugh–Nagumo system has been done in [10] where an optimal control result is obtained. In this paper we study the approximate controllability for the system (1), with δ = 0. The following assumptions …