Closed-loop control of a reaction-diffusion system

A system of a parabolic partial differential equation coupled with ordinary differential inclusions that arises from a closed-loop control problem for a thermodynamic process governed by the Allen-Cahn diffusion reaction model is studied. A feedback law for the closed-loop control is proposed and implemented in the case of a finite number of control devices located inside the process domain basing on the process dynamics observed at a finite number of measurement points. The existence of solutions to the discussed system of differential equations is proved with the use of a generalization of the Kakutani fixed point theorem.

💡 Research Summary

The paper investigates a closed‑loop control problem for a thermodynamic process described by the Allen‑Cahn reaction‑diffusion equation. The authors consider a parabolic PDE coupled with a finite number of ordinary differential inclusions that model internal control devices. The control devices are placed inside the spatial domain and act through an additive term g(x,t;κ(t)) that is a linear combination of the device signals κ_j(t). Measurements are taken at a finite set of interior points x*_k, and the control law feeds back the deviation of the measured state from prescribed target values.

Mathematically, the system reads
u_t – Δu = f(u) + g(x,t;κ(t)) in Ω×(0,T),
β_j ·κ̇_j + κ_j ∈ W_j(t, u(x*_1,t),…,u(x*_n,t)) for j=1,…,m,
with homogeneous Neumann boundary conditions and given initial data. The nonlinear reaction term f satisfies linear growth and local Lipschitz conditions; the multivalued maps W_j are non‑empty, closed, convex, bounded, and upper semicontinuous. The control term g is assumed to belong to L²(0,T;L^∞(Ω)) and to be Lipschitz with respect to κ.

The authors define a weak solution concept for the PDE and a Carathéodory solution concept for the inclusions. They then construct three operators:

1. R maps a control κ to the corresponding PDE solution u and extracts the values u(x*_k,·).
2. Q assigns to a given measurement trajectory f(t) the set of admissible power supplies v(t) that lie in W(t,f(t)).
3. P solves the linear ODE β_j ·κ̇_j + κ_j = v_j(t) for any bounded v, producing a control κ in the Sobolev space W^{1,∞}(0,T).

The composition Φ = P ∘ Q ∘ R acts from a compact convex subset M_S of C(