Local exact controllability of a reaction-diffusion system
The system consists of the two coupled parabolic equations \[ \begin{aligned} {\partial y(t, x) \over \partial t} &- k_1 \Delta y(t, x) + \alpha a(x) y(t, x) z(t, x) = f(x) + u(t, x),\\ {\partial z(t, x) \over \partial t} &- k_2 \Delta z(t, x) + \beta a(x) y(t, x) z(t, x) = g(x) + v(t, x) \end{aligned} \] in a bounded \(n\)-dimensional domain \(\Omega,\) \((n = 1, 2, 3)\) with Neumann boundary condition; the control functions \(u(t, x),\) \(v(t, x)\) have support \(\subseteq \omega \subset \Omega.\) Under suitable assumptions, the authors show that the system is locally null controllable. The tools are Carleman estimates for the backwards adjoint linearized system and the Schauder-Kakutani fixed point theorem.
scientific article; zbMATH DE number 1832832