Advanced engineering analysis : the calculus of variations and functional analysis with applications in mechanics

Content: 1. Basic calculus of variations. 1.1. Introduction. 1.2. Euler's equation for the simplest problem. 1.3. Properties of extremals of the simplest functional. 1.4. Ritz's method. 1.5. Natural boundary conditions. 1.6. Extensions to more general functionals. 1.7. Functionals depending on functions in many variables. 1.8. A functional with integrand depending on partial derivatives of higher order. 1.9. The first variation. 1.10. Isoperimetric problems. 1.11. General form of the first variation. 1.12. Movable ends of extremals. 1.13. Broken extremals: Weierstrass-Erdmann conditions and related problems. 1.14. Sufficient conditions for minimum. 1.15. Exercises --
2. Applications of the calculus of variations in mechanics. 2.1. Elementary problems for elastic structures. 2.2. Some extremal principles of mechanics. 2.3. Conservation laws. 2.4. Conservation laws and Noether's theorem. 2.5. Functionals depending on higher derivatives of y. 2.6. Noether's theorem, general case. 2.7. Generalizations. 2.8. Exercises --
3. Elements of optimal control theory. 3.1. A variational problem as an optimal control problem. 3.2. General problem of optimal control. 3.3. Simplest problem of optimal control. 3.4. Fundamental solution of a linear ordinary differential equation. 3.5. The simplest problem, continued. 3.6. Pontryagin's maximum principle for the simplest problem. 3.7. Some mathematical preliminaries. 3.8. General terminal control problem. 3.9. Pontryagin's maximum principle for the terminal optimal problem. 3.10. Generalization of the terminal control problem. 3.11. Small variations of control function for terminal control problem. 3.12. A discrete version of small variations of control function for generalized terminal control problem. 3.13. Optimal time control problems. 3.14. Final remarks on control problems. 3.15. Exercises. 4. Functional analysis. 4.1. A normed space as a metric space. 4.2. Dimension of a linear space and separability. 4.3. Cauchy sequences and Banach spaces. 4.4. The completion theorem. 4.5. L[symbol] spaces and the Lebesgue integral. 4.6. Sobolev spaces. 4.7. Compactness. 4.8. Inner product spaces, Hilbert spaces. 4.9. Operators and functionals. 4.10. Contraction mapping principle. 4.11. Some approximation theory. 4.12 Orthogonal decomposition of a Hilbert space and the Riesz representation theorem. 4.13. Basis, Gram-Schmidt procedure, and Fourier series in Hilbert space. 4.14. Weak convergence. 4.15. Adjoint and self-adjoint operators. 4.16. Compact operators. 4.17. Closed operators. 4.18. On the Sobolev imbedding theorem. 4.19. Some energy spaces in mechanics. 4.20. Introduction to spectral concepts. 4.21. The Fredholm theory in Hilbert spaces. 4.22. Exercises --
5. Applications of functional analysis in mechanics. 5.1. Some mechanics problems from the standpoint of the calculus of variations
 the virtual work principle. 5.2. Generalized solution of the equilibrium problem for a clamped rod with springs. 5.3. Equilibrium problem for a clamped membrane and its generalized solution. 5.4. Equilibrium of a free membrane. 5.5. Some other equilibrium problems of linear mechanics. 5.6. The Ritz and Bubnov-Galerkin methods. 5.7. The Hamilton-Ostrogradski principle and generalized setup of dynamical problems in classical mechanics. 5.8. Generalized setup of dynamic problem for membrane. 5.9. Other dynamic problems of linear mechanics. 5.10. The Fourier method. 5.11. An eigenfrequency boundary value problem arising in linear mechanics. 5.12. The spectral theorem. 5.13. The Fourier method, continued. 5.14. Equilibrium of a von Karman plate. 5.15. A unilateral problem. 5.16. Exercises.