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

State Transition Tensors for Continuous-Thrust Control of Three-Body Relative Motion

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.

Optimal control problems for the reaction–diffusion–convection equation with variable coefficients

We study a quantum scalar field, with general mass and coupling to the scalar curvature, propagating on three-dimensional global anti-de Sitter space-time. We determine the vacuum and thermal expectation values of the square of the field, also known as the vacuum polarisation (VP). We consider values of the scalar field mass and coupling for which there is a choice of boundary conditions giving well-posed classical dynamics. We apply Dirichlet, Neumann and Robin (mixed) boundary conditions to the field at the space-time boundary. We find finite values of the VP when the parameter governing the Robin boundary conditions is below a certain critical value. For all couplings, the vacuum expectation values of the VP with either Neumann or Dirichlet boundary conditions are constant and respect the maximal symmetry of the background space-time. However, this is not the case for Robin boundary conditions, when both the vacuum and thermal expectation values depend on the space-time location. At the space-time boundary, we find that both the vacuum and thermal expectation values of the VP with Robin boundary conditions converge to the result when Neumann boundary conditions are applied, except in the case of Dirichlet boundary conditions.

Dynamic programming approach to multiple objective control problem having deterministic or fuzzy goals

We study Dirichlet boundary control problem for semilinear parabolic equations, with pointwise state constraints. By penalizing the Dirichlet boundary condition, we define a family of control problems with Robin boundary conditions. We study the asymptotic behavior of solutions of these penalized problems, when the penalty parameter tends to infinity. We also prove that optimality conditions for the Dirichlet boundary control problem can be obtained by passage to the limit in the optimality conditions of the penalized problem.

We compute the vacuum polarization for a massless, conformally coupled scalar field on the covering space of global, four-dimensional, anti-de Sitter space-time. Since anti-de Sitter space is not globally hyperbolic, boundary conditions must be applied to the scalar field. We consider general Robin (mixed) boundary conditions for which the classical evolution of the field is well-defined and stable. The vacuum expectation value of the square of the field is not constant unless either Dirichlet or Neumann boundary conditions are applied. We also compute the thermal expectation value of the square of the field. For Dirichlet boundary conditions, both thermal and vacuum expectation values approach the same well-known limit on the space-time boundary. For all other Robin boundary conditions (including Neumann boundary conditions), the vacuum and thermal expectation values have the same limit on the space-time boundary, but this limit does not equal that in the Dirichlet case.

This work is concerned with the maximum principles for optimal control problems governed by 3-dimensional Navier--Stokes equations. Some types of state constraints (time variables) are considered.

This thesis deals with the study of optimal control problems for the incompressible Magnetohydrodynamics (MHD) equations. Particular attention to these problems arises from several applications in science and engineering, such as fission nuclear reactors with liquid metal coolant and aluminum casting in metallurgy. In such applications it is of great interest to achieve the control on the fluid state variables through the action of the magnetic Lorentz force. In this thesis we investigate a class of boundary optimal control problems, in which the flow is controlled through the boundary conditions of the magnetic field. Due to their complexity, these problems present various challenges in the definition of an adequate solution approach, both from a theoretical and from a computational point of view. In this thesis we propose a new boundary control approach, based on lifting functions of the boundary conditions, which yields both theoretical and numerical advantages. With the introduction of lifting functions, boundary control problems can be formulated as extended distributed problems. We consider a systematic mathematical formulation of these problems in terms of the minimization of a cost functional constrained by the MHD equations. The existence of a solution to the flow equations and to the optimal control problem are shown. The Lagrange multiplier technique is used to derive an optimality system from which candidate solutions for the control problem can be obtained. In order to achieve the numerical solution of this system, a finite element approximation is considered for the discretization together with an appropriate gradient-type algorithm. A finite element object-oriented library has been developed to obtain a parallel and multigrid computational implementation of the optimality system based on a multiphysics approach. Numerical results of two- and three-dimensional computations show that a possible minimum for the control problem can be computed in a robust and accurate manner.

Analysis of mass transfer parameters (changes in mass flux, diffusion coefficient and mass transfer coefficient) during baking of cookies

The global solvability of the inhomogeneous mixed boundary value problem and control problems for the reaction–diffusion–convection equation are proved in the case when the reaction coefficient nonlinearly depends on the concentration. The maximum and minimum principles are established for the solution of the boundary value problem. The optimality systems are derived and the local stability estimates of optimal solutions are established for control problems with specific reaction coefficients

Global solvability of a boundary value problem for a generalized Boussinesq model is proved in the case, when reaction coefficient depends nonlinearly on concentration of substance. Maximum principle is stated for substance’s concentration. Solvability of control problem is proved, when the role of controls is played by diffusion and mass exchange coefficients from the equations and from the boundary conditions of the model. For a considered multiplicative control problem, optimality systems are obtained. On the base of the analysis of these systems for particular reaction coefficients and cost functionals, local stability estimates are deduced for optimal solutions.

A class of boundary-distributed linear control systems in Banach spaces is studied. A maximum principle for a convex control problem associated with such systems is obtained.

The power-series method, i.e., a finite analytic approach based on power-series expansion, was applied to transpiration cooling problems, and the stability and accuracy of the method were evaluated. Stability analysis using von Neumann's method showed that the power-series method was stable for transpiration-cooling problems on the condition that Δx ≥ gΔt, where g is a mass flow rate parameter. The solutions obtained with the power-series method for typical problems were compared with those obtained with the fully implicit finite-difference method. The comparisons revealed that the power-series method yielded more accurate solutions for problems with Robin and Neumann boundary conditions, but less accurate solutions for problems with Robin and Dirichlet boundary conditions. For problems with Robin and Robin boundary conditions, the accuracy of the power-series method depended on the value of the mass flow rate parameter g.

Global existence is proved for the solution, of a Riccati differential equation connected with the synthesis of a boundary control problem governed by parabolic partial differential equations.