Boundary control problem for the reaction–advection–diffusion equation with a modulus discontinuity of advection

We consider a periodic problem for a singularly perturbed parabolic reaction–diffusion–advection equation of the Burgers type with the modulus advection; it has a solution in the form of a moving front. We formulate conditions for the existence of such a solution and construct its asymptotic approximation. We pose a control problem where the required front propagation law is implemented by a specially chosen boundary condition. We construct an asymptotic solution of the boundary control problem. Using the asymptotic method of differential inequalities, we estimate the accuracy of the solution of the control problem. We propose an original numerical algorithm for solving singularly perturbed problems involving the modulus advection.