Extremum problems of boundary control for a stationary thermal convection model

Abstract

151 Recently, the control theory of thermal and hydro� dynamic fields in continuous media has been inten� sively developed. The mathematical description of such problems includes three components: a goal, the control mechanisms used to achieve this goal, and constraints imposed on the state and controls of the system under study. The constraints are specified by the equations of the used continuous medium model together with boundary conditions set on the bound� ary of the domain, while the desired goal is achieved by minimizing a certain cost functional. Control problems and inverse extremum problems for stationary models of heat and mass transfer have been theoretically studied, for example, in [1–8], where the solvability of optimization problems was analyzed and optimality systems describing necessary extremum conditions were derived and examined. In [6–8], based on an analysis of an optimality system, sufficient conditions for the uniqueness and stability of solutions to control problems were established in spe� cial cases corresponding to purely hydrodynamic or temperature cost functionals and controls. The con� trol of viscous flows by applying heat sources and the control of thermal processes by applying hydrody� namic sources have been investigated to a lesser degree. At the same time, computations show that effective mechanisms for the control of viscous flows can be found in this direction. The goal of this paper is to study such control prob� lems for the Oberbeck–Boussinesq thermal convec� tion model. First, we formulate the general control problem for this model and present a theorem provid� ing sufficient conditions on the initial data under which the solution of the control problem is unique and stable with respect to small perturbations in the cost functional and the boundary function involved in the Dirichlet boundary condition for velocity. Next, we describe a numerical solution algorithm for the control problem based on Newton’s method and dis� cuss numerical results. They confirm that the algo� rithm is highly effective in a wide range of the basic parameters of the boundary value problem.