Boundary control of fronts in a Burgers-type equation with modular adhesion and periodic amplification

V.T. Volkov, N. N. Nefedov

Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia

Abstract: We consider a singularly perturbed periodic problem for a Burgers-type equation with modular advection and periodic linear amplification. We obtain conditions for the existence, uniqueness, and asymptotic stability in the sense of Lyapunov of a periodic solution with an interior transition layer and construct its asymptotic approximation. The asymptotics of the solution is used to determine boundary conditions ensuring the implementation of a prescribed mode of the front motion, i.e., the boundary control problem. We also formulate the notion of an asymptotic solution of the boundary control problem and obtain sufficient conditions for the existence of the required periodic mode of the front motion.

Keywords: singularly perturbed parabolic equations, periodic problem, reaction–diffusion–advection equations, contrast structure, interior layer, moving front, asymptotic methods, differential inequalities, asymptotic stability in the sense of Lyapunov, Burgers equations with modular advection, boundary control problem, asymptotic solution of boundary control problem.

Received: 24.01.2022
Revised: 26.03.2022

DOI: 10.4213/tmf10258

 English version: Theoretical and Mathematical Physics, 2022, 212:2, 1044–1052

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