Analysis of a multiplicative control problem for a nonlinear reaction–diffusion–convection equation

Abstract The control problem for a nonlinear reaction–diffusion–convection equation is studied. The role of multiplicative controls is played by the diffusion coefficient, by the mass transfer coefficient in the Robin boundary condition and also by the velocity vector. Boundary and distributed controls are also used. The solvability of the extremum problem is proved under minimal conditions on multiplicative controls. For a specified reaction coefficient optimality systems are derived for control problems. On the basis of the analysis of these systems the bang–bang principle for a distributed control is established; additionally, the local stability estimates of optimal solutions are derived with respect to small perturbations of both cost functionals and of one of specified functions from the boundary value problem.