Operator Theory

1 Basic Definitions and Auxiliary Statements.- 1.1 Sets, functions, real numbers.- 1.1.1 Notations and definitions.- 1.1.2 Real numbers.- 1.2 Topological, metric, and normed spaces.- 1.2.1 General notions.- 1.2.2 Metric spaces.- 1.2.3 Normed vector spaces.- 1.3 Continuous functions and compact spaces.- 1.3.1 Continuous and semicontinuous mappings.- 1.3.2 Compact spaces.- 1.3.3 Continuous functions on compact spaces.- 1.4 Maximum function and its properties.- 1.4.1 Discrete maximum function.- 1.4.2 General maximum function.- 1.5 Hilbert space.- 1.5.1 Basic definitions and properties.- 1.5.2 Compact and selfadjoint operators in a Hilbert space.- 1.5.3 Theorem on continuity of a spectrum.- 1.5.4 Embedding of a Hilbert space in its dual.- 1.5.5 Scales of Hilbert spaces and compact embedding.- 1.6 Functional spaces that are used in the investigation of boundary value and optimal control problems.- 1.6.1 Spaces of continuously differentiable functions.- 1.6.2 Spaces of integrable functions.- 1.6.3 Test and generalized functions.- 1.6.4 Sobolev spaces.- 1.7 Inequalities of coerciveness.- 1.7.1 Coercive systems of operators.- 1.7.2 Korn's inequality.- 1.8 Theorem on the continuity of solutions of functional equations.- 1.9 Differentiation in Banach spaces and the implicit function theorem.- 1.9.1 Frechet derivative and its properties.- 1.9.2 Implicit function.- 1.9.3 The Gateaux derivative and its connection with the Frechet derivative.- 1.10 Differentiation of the norm in the space Wpm(?).- 1.10.1 Auxiliary statement.- 1.10.2 Theorem on differentiability.- 1.11 Differentiation of eigenvalues.- 1.11.1 The eigenvalue problem.- 1.11.2 Differentiation of an operator-valued function.- 1.11.3 Eigenspaces and projections.- 1.11.4 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.- 1.14.1 Diffeomorphisms and the relations between the derivatives.- 1.14.2 Sequential Frechet derivatives and partial derivatives of a composite function.- 1.14.3 Theorem on the invariance of Sobolev spaces.- 1.14.4 Transformation of derivatives under the change of variables.- 2 Optimal Control by Coefficients in Elliptic Systems.- 2.1 Direct problem.- 2.1.1 Coercive forms and operators.- 2.1.2 Boundary value problem.- 2.2 Optimal control problem.- 2.2.1 Nonregular control.- 2.2.2 Regular control.- 2.2.3 Regular problem and necessary conditions of optimality.- 2.2.4 Nonsmooth (discontinuous) control.- 2.2.5 Some remarks on the use of regular and discontinuous controls.- 2.3 The finite-dimensional problem.- 2.4 The finite-dimensional problem (another approach).- 2.4.1 The set U(t).- 2.4.2 Approximate solution of the problem (2.2.22).- 2.4.3 Approximate solution of the optimal control problem when the set ?ad is empty.- 2.4.4 On the computation of the functional h ? ?k(h,uh).- 2.4.5 Calculation and use of the Frechet derivative of the functional h ? ?ma(h,uh).- 2.5 Spectral problem.- 2.5.1 Eigenvalue problem.- 2.5.2 On the continuity of the spectrum.- 2.6 Optimization of the spectrum.- 2.6.1 Formulation of the problem and the existence theorem.- 2.6.2 Finite-dimensional approximation of the optimal control problem.- 2.6.3 Computation of eigenvalues.- 2.7 Control under restrictions on the spectrum.- 2.7.1 Optimal control problem.- 2.7.2 Approximate solution of the problem (2.7.7).- 2.7.3 Second method of approximate solution of the problem (2.7.7).- 2.7.4 Differentiation of the functionals h ? Aiu(h) and necessary conditions of optimality.- 2.8 The basic optimal control problem.- 2.8.1 Setting of the problem. Existence theorem.- 2.8.2 Approximate solution of the problem (2.8.6).- 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.- 2.10.1 Setting of the problem.- 2.10.2 The existence theorem.- 2.11 The general control problem.- 2.11.1 Bilinear form aq and the corresponding equation.- 2.11.2 Bilinear form br and the spectral problem.- 2.11.3 Basic control problem.- 2.11.4 Application of the basic control problem (combined problem).- 2.12 Optimization by the shape of domain and by operators.- 2.12.1 Domains and bilinear forms.- 2.12.2 Optimization problem connected with solution of an operator equation.- 2.12.3 Eigenvalue optimization problem.- 2.12.4 Some realizations of the spaces Ml and Nl.- 2.13 Optimization problems with smooth solutions of state equations.- 2.13.1 Systems of elliptic equations.- 2.13.2 Elliptic problems in domains and in a fixed domain.- 2.13.3 The problem of domain shape optimization.- 2.13.4 Approximate solution of the direct problem ensuring convergence in the norm of a space of smooth functions.- 3 Control by the Right-hand Sides in Elliptic Problems.- 3.1 On the minimum of nonlinear functionals.- 3.1.1 Setting of the problem. Auxiliary statements.- 3.1.2 The existence theorem.- 3.1.3 Characterization of a minimizing element.- 3.1.4 Functionals continuous in the weak topology.- 3.2 Approximate solution of the minimization problem.- 3.2.1 Inner point lemma.- 3.2.2 Finite-dimensional problem.- 3.3 Control by the right-hand side in elliptic problems provided the goal functional is quadratic.- 3.3.1 Setting of the problem.- 3.3.2 Existence of a solution. Optimality conditions.- 3.3.3 An example of a system described by the Dirichlet problem.- 3.4 Minimax control problems.- 3.5 Control of systems whose state is described by variational inequalities.- 3.5.1 Setting of the problem.- 3.5.2 The existence theorem.- 3.5.3 An example of control of a system described by a variational inequality.- 4 Direct Problems for Plates and Shells.- 4.1 Bending and free oscillations of thin plates.- 4.1.1 Basic relations of the theory of bending of thin plates.- 4.1.2 Orthotropic plates.- 4.1.3 Bilinear form corresponding to the strain energy of the plate.- 4.1.4 Problem of bending of a plate.- 4.1.5 Problem of free oscillations of a plate.- 4.2 Problem of stability of a thin plate.- 4.2.1 Stored energy of a plate.- 4.2.2 Conditions of stationarity.- 4.2.3 Auxiliary statements.- 4.2.4 Transformation of the problem (4.2.27), (4.2.28).- 4.2.5 Stability of a plate and bifurcation.- 4.2.6 An example of nonexistence of stable solutions.- 4.3 Model of the three-layered plate ignoring shears in the middle layer.- 4.3.1 Basic relations.- 4.3.2 Problems of the bending and of the free flexural oscillations.- 4.4 Model of the three-layered plate accounting for shears in the middle layer.- 4.4.1 Basic relations.- 4.4.2 Bilinear form corresponding to the three-layered plate.- 4.4.3 Bending of the three-layered plate.- 4.4.4 Natural oscillations of three-layered plate.- 4.5 Basic relations of the shell theory.- 4.6 Shells of revolution.- 4.6.1 Deformations and functional spaces.- 4.6.2 The bilinear form ah.- 4.6.3 The subspace of functions with zero-point strain energy.- 4.7 Shallow shells.- 4.8 Problems of statics of shells.- 4.9 Free oscillations of a shell.- 4.10 Problem of shell stability.- 4.10.1 On some approaches to stability problems.- 4.10.2 Reducing of the stability problem to the eigenvalue problem.- 4.10.3 Spectral problem (4.10.12).- 4.11 Finite shear model of a shell.- 4.11.1 Strain energy of an elastic shell.- 4.11.2 Shallow shell.- 4.11.3 A relation between the Kirchhoff and Timoshenko models of shell.- 4.12 Laminated shells.- 4.12.1 The strain energy of a laminated shell.- 4.12.2 Shell of revolution.- 4.12.3 Shallow shells.- 5 Optimization of Deformable Solids.- 5.1 Settings of optimization problems for plates and shells.- 5.1.1 Goal functional and a function of control.- 5.1.2 Restrictions.- 5.2 Approximate solution of direct and optimization problems for plates and shells.- 5.2.1 Direct problems and spline functions.- 5.2.2 The spaces Vm for plates.- 5.2.3 The spaces Vm for shells.- 5.2.4 Direct problems for nonfastened plates and shells.- 5.2.5 Solution of optimization problems.- 5.3 Optimization problems for plates (control by the function of the thickness).- 5.3.1 Optimization under restrictions on strength.- 5.3.2 Stability optimization problem.- 5.3.3 Optimization of frequencies of free oscillations.- 5.3.4 Combined optimization problem and optimization for a class of loads.- 5.4 Optimization problems for shells (control by functions of midsurface and thickness).- 5.4.1 Problem of optimization of a shell of revolution with respect to strength.- 5.4.2 Optimization according to the stability of a cylindrical shell subject to a hydrostatic compressive load.- 5.5 Control by the shape of a hole and by the function of thickness for a shallow shell.- 5.5.1 Problem of optimization according to strength.- 5.5.2 Approximate solution of the optimization and direct problems.- 5.5.3 Problem of optimization of eigenvalues.- 5.5.4 Approximate solution of the eigenvalue problem.- 5.6 Control by the load for plates and shells.- 5.6.1 General problem of control by the load.- 5.6.2 Optimization problems for plates.- 5.7 Optimization of structures of composite materials.- 5.7.1 Concept of a composite material.- 5.7.2 Homogenization (averaging) of a periodical structure based on G-convergence.- 5.7.3 Effective elasticity characteristics of granule and fiber reinforced composites.- 5.7.4 Optimization of the effective elasticity constants of a composite.- 5.7.5 Optimization of a granule reinforced composite.- 5.7.6 Optimization of composite laminate shells.- 5.7.7 Optimization of the composite structure.- 5.8 Optimization of laminate composite covers according to mechanical and radio engineering characteristics.- 5.8.1 Propagation of electromagnetic waves through a laminated medium.- 5.8.2 Optimization problems.- 5.9 Shape optimization of a two-dimensional elastic body.- 5.9.1 Sets of controls and domains in the optimization problem.- 5.9.2 Problems of elasticity in domains.- 5.9.3 The optimization problem.- 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.- 5.11.1 Optimization problem for an elastic solid.- 5.11.2 Spaces and operators on ?/2??, auxiliary statements.- 5.11.3 Optimization problem on ?/2??.- 5.12 Optimization of the residual stresses in an elastoplastic body.- 5.12.1 Force and thermal loading of a nonlinear elastoplastic body.- 5.12.2 Residual stresses and deformations.- 5.12.3 Temperature pattern in a medium.- 5.12.4 Optimization problem.- 6 Optimization Problems for Steady Flows of Viscous and Nonlinear Viscous Fluids.- 6.1 Problem of steady flow of a nonlinear viscous fluid.- 6.1.1 Basic equations and assumptions.- 6.1.2 Formulation of the problem.- 6.1.3 Existence theorem.- 6.2 Theorem on continuity.- 6.3 Continuity with respect to the shape of the domain.- 6.3.1 Formulation of the problem.- 6.3.2 Lemmas on operators $$ {tilde L_q}and{tilde B_q}$$.- 6.3.3 Theorem on continuity.- 6.4 Control of fluid flows by perforated walls and computation of the function of filtration.- 6.4.1 The problem of flow in a circular cylinder and the function of filtration.- 6.4.2 The passage factor for the power model.- 6.4.3 Control of the surface forces at the inlet by the perforated wall.- 6.5 The flow in a canal with a perforated wall placed inside.- 6.5.1 Basic equations.- 6.5.2 Generalized solution of the problem.- 6.6 Optimization by the functions of surface forces and filtration.- 6.6.1 Formulation of the problem and the existence theorem.- 6.6.2 On the differentiability of the function T?(v(T), p(T)).- 6.6.3 Differentiability of the functionals ?iand necessary optimality conditions.- 6.7 Problems of the optimal shape of a canal.- 6.7.1 Set of controls and diffeomorphisms.- 6.7.2 Optimization problems.- 6.8 A problem of the optimal shape of a hydrofoil.- 6.8.1 State equation for a moving hydrofoil.- 6.8.2 Fixed-domain problem and Frechet differentiability of the functionals.- 6.8.3 Optimization problem.- 6.9 Direct and optimization problems with consideration for the inertia forces.- 6.9.1 Setting and solution of the direct problem.- 6.9.2 Approximation of the problem (6.9.10)-(6.9.12).- 6.9.3 Some remarks on models, optimization problems, and existence results.