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The calculus of variations and functional analysis : with optimal control and applications in mechanics /
This is a book for those who want to understand the main ideas in the theory of optimal problems. It provides a good introduction to classical topics (under the heading of "the calculus of variations") and more modern topics (under the heading of "optimal control"). It employs th...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Otros Autores: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Singapore ; River Edge, N.J. :
World Scientific,
©2003.
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| Colección: | Series on stability, vibration, and control of systems.
v. 12. |
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- 1. Basic calculus of variations. 1.1. Introduction. 1.2. Euler's equation for the simplest problem. 1.3. Some properties of extremals of the simplest functional. 1.4. Ritz's method. 1.5. Natural boundary conditions. 1.6. Some 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. Weierstrass-Erdmann conditions and related problems. 1.14. Sufficient conditions for minimum. 1.15. Exercises
- 2. Elements of optimal control theory. 2.1. A variational problem as a problem of optimal control. 2.2. General problem of optimal control. 2.3. Simplest problem of optimal control. 2.4. Fundamental solution of a linear ordinary differential equation. 2.5. The simplest problem, continued. 2.6. Pontryagin's maximum principle for the simplest problem. 2.7. Some mathematical preliminaries. 2.8. General terminal control problem. 2.9. Pontryagin's maximum principle for the terminal optimal problem. 2.10. Generalization of the terminal control problem. 2.11. Small variations of control function for terminal control problem. 2.12. A discrete version of small variations of control function for generalized terminal control problem. 2.13. Optimal time control problems. 2.14. Final remarks on control problems. 2.15. Exercises.
- 3. Functional analysis. 3.1. A normed space as a metric space. 3.2. Dimension of a linear space and separability. 3.3. Cauchy sequences and Banach spaces. 3.4. The completion theorem. 3.5. Contraction mapping principle. 3.6. L[symbol] spaces and the Lebesgue integral. 3.7. Sobolev spaces. 3.8. Compactness. 3.9. Inner product spaces, Hilbert spaces. 3.10. Some energy spaces in mechanics. 3.11. Operators and functionals. 3.12. Some approximation theory. 3.13. Orthogonal decomposition of a Hilbert space and the Riesz representation theorem. 3.14. Basis, Gram-Schmidt procedure, Fourier series in Hilbert space. 3.15. Weak convergence. 3.16. Adjoint and self-adjoint operators. 3.17. Compact operators. 3.18. Closed operators. 3.19. Introduction to spectral concepts. 3.20. The Fredholm theory in Hilbert spaces. 3.21. Exercises
- 4. Some applications in mechanics. 4.1. Some problems of mechanics from the viewpoint of the calculus of variations; the virtual work principle. 4.2. Equilibrium problem for a clamped membrane and its generalized solution. 4.3. Equilibrium of a free membrane. 4.4. Some other problems of equilibrium of linear mechanics. 4.5. The Ritz and Bubnov-Galerkin methods. 4.6. The Hamilton-Ostrogradskij principle and the generalized setup of dynamical problems of classical mechanics. 4.7. Generalized setup of dynamic problems for a membrane. 4.8. Other dynamic problems of linear mechanics. 4.9. The Fourier method. 4.10. An eigenfrequency boundary value problem arising in linear mechanics. 4.11. The spectral theorem. 4.12. The Fourier method, continued. 4.13. Equilibrium of a von Kármán plate. 4.14. A unilateral problem. 4.15. Exercises.