Optimization in Elliptic Problems with Applications to Mechanics of Deformable Bodies and Fluid Mech

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 Fréchet derivative and its properties.- 1.9.2 Implicit function.- 1.9.3 The Gâteaux derivative and its connection with the Fréchet 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 Fréchet 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 Fréchet 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 ? Aiµ(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.