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 ...
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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.
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