Calculus of variations
Ects : 5
Enseignant responsable :
Volume horaire : 39Description du contenu de l'enseignement :
- Introduction to variational problems in infinite dimension
- Classical examples: geodesics, brachistochrone, Ramsey growth model, Bolza problem
- Relation to optimal control
- Euler-Lagrange equations : formal derivation, rigorous derivation when there is a smooth solution and resolution
- Existence of solutions : direct method in the Calculus of Variations.
- Direct method, weak convergence and weak semicontinuity
- Weierstrass theorem
- Weak and weak-* topology, relation with convexity, compactness
- Weak lower semicontinuity of integral functionals of order 0
- Convex analysis and optimization
- Optimization of extended-real-valued functions
- Convex sets: geometric and topological properties, Hahn–Banach
- Convex functions: definition, regularity, subgradients and subdifferentials, convex conjugate, convex duality
- Integral functionals of order 1
- Sobolev spaces (in dimension 1)
- Continuous and compact embeddings
- Semicontinuity of integral functionals of order 1
- Euler-Lagrange equations : weak formulation, regularity a posteriori
- Convex duality and applications
Compétence à acquérir :
Introduction to the calculus of variations, with a focus on convex and one-dimensional variational problems.