This book is intended to be both a thorough introduction to contemporary research in optimization theory for elliptic systems with its numerous applications and a textbook at the undergraduate and graduate level for courses in pure or applied mathematics or in continuum mechanics. Various processes of modern technology and production are described by el­ liptic partial differential equations. Optimization of these processes reduces to op­ timization problems for elliptic systems. The numerical solution of such problems is associated with the solution of the following questions. 1. The setting of the optimization problem ensuring the existence of a solution on a set of admissible controls, which is a subset of some infinite-dimensional vector space. 2. Reduction of the infinite-dimensional optimization problem to a sequence of finite-dimensional problems such that the solutions of the finite-dimensional problems converge, in a sense, to the solution of the infinite-dimensional problem.3. Numerical solution of the finite-dimensional problems.

Table of Contents:
1 Basic Definitions and Auxiliary Statements.- 1.1 Sets, functions, real numbers.- 1.2 Topological, metric, and normed spaces.- 1.3 Continuous functions and compact spaces.- 1.4 Maximum function and its properties.- 1.5 Hilbert space.- 1.6 Functional spaces that are used in the investigation of boundary value and optimal control problems.- 1.7 Inequalities of coerciveness.- 1.8 Theorem on the continuity of solutions of functional equations.- 1.9 Differentiation in Banach spaces and the implicit function theorem.- 1.10 Differentiation of the norm in the space Wpm(?).- 1.11 Differentiation of eigenvalues.- 1.12 The Lagrange principle in smooth extremum problems.- 1.13 G-convergence and G-closedness of linear operators.- 1.14 Diffeomorphisms and invariance of Sobolev spaces with respect to diffeomorphisms.- 2 Optimal Control by Coefficients in Elliptic Systems.- 2.1 Direct problem.- 2.2 Optimal control problem.- 2.3 The finite-dimensional problem.- 2.4 The finite-dimensional problem (another approach).- 2.5 Spectral problem.- 2.6 Optimization of the spectrum.- 2.7 Control under restrictions on the spectrum.- 2.8 The basic optimal control problem.- 2.9 The combined problem.- 2.10 Optimal control problem for the case when the state of the system is characterized by a set of functions.- 2.11 The general control problem.- 2.12 Optimization by the shape of domain and by operators.- 2.13 Optimization problems with smooth solutions of state equations.- 3 Control by the Right-hand Sides in Elliptic Problems.- 3.1 On the minimum of nonlinear functionals.- 3.2 Approximate solution of the minimization problem.- 3.3 Control by the right-hand side in elliptic problems provided the goal functional is quadratic.- 3.4 Minimax control problems.- 3.5 Control of systems whose state is described by variational inequalities.- 4 Direct Problems for Plates and Shells.- 4.1 Bending and free oscillations of thin plates.- 4.2 Problem of stability of a thin plate.- 4.3 Model of the three-layeredplate ignoring shears in the middle layer.- 4.4 Model of the three-layered plate accounting for shears in the middle layer.- 4.5 Basic relations of the shell theory.- 4.6 Shells of revolution.- 4.7 Shallow shells.- 4.8 Problems of statics of shells.- 4.9 Free oscillations of a shell.- 4.10 Problem of shell stability.- 4.11 Finite shear model of a shell.- 4.12 Laminated shells.- 5 Optimization of Deformable Solids.- 5.1 Settings of optimization problems for plates and shells.- 5.2 Approximate solution of direct and optimization problems for plates and shells.- 5.3 Optimization problems for plates (control by the function of the thickness).- 5.4 Optimization problems for shells (control by functions of midsurface and thickness).- 5.5 Control by the shape of a hole and by the function of thickness for a shallow shell.- 5.6 Control by the load for plates and shells.- 5.7 Optimization of structures of composite materials.- 5.8 Optimization of laminate composite covers according to mechanical and radio engineering characteristics.- 5.9 Shape optimization of a two-dimensional elastic body.- 5.10 Optimization of the internal boundary of a two-dimensional elastic body.- 5.11 Optimization problems on manifolds and shape optimization of elastic solids.- 5.12 Optimization of the residual stresses in an elastoplastic body.- 6 Optimization Problems for Steady Flows of Viscous and Nonlinear Viscous Fluids.- 6.1 Problem of steady flow of a nonlinear viscous fluid.- 6.2 Theorem on continuity.- 6.3 Continuity with respect to the shape of the domain.- 6.4 Control of fluid flows by perforated walls and computation of the function of filtration.- 6.5 The flow in a canal with a perforated wall placed inside.- 6.6 Optimization by the functions of surface forces and filtration.- 6.7 Problems of the optimal shape of a canal.- 6.8 A problem of the optimal shape of a hydrofoil.- 6.9 Direct and optimization problems with consideration for the inertia forces.