Multiplicative Control Problem for a Nonlinear Reaction–Diffusion Model

Abstract

The paper studies a multiplicative control problem for the reaction–diffusion equation in which the reaction coefficient nonlinearly depends on the substance concentration, as well as on spatial variables. The role of multiplicative controls is played by the coefficients of diffusion and mass transfer. The solvability of the extremum problem is proved, and optimality systems are derived for a specific reaction coefficient. Based on the analysis of these systems, the relay property of multiplicative and distributed controls is established, and estimates of the local stability of optimal solutions to small perturbations of both the quality functionals and one of the given functions of the boundary value problem are derived.