I semester

Introduction:

Homogenization in studying and control of complex systems.

Part 1. Problems with one independent variable.

One dimensional homogenization.

• Canonical form, averaging.
• Examples. effective conductivity, speed of sound, etc.

Introduction to optimal control

• Control theory: variables, controls, functionals. Examples.
• Canonical form and Pontriagin's maximum principle.
• Chattering control and averaging.
• Dynamical programming.

Homogenization of elliptic PDE

• Equation of second order.
• Homogenization by multi-scale expansions. Effective properties.
• Bounds for effective properties.
• Elasticity equations. Homogenization.
• Non-linear problems.

Control of systems described by elliptic PDE.

• Examples of optimal control: variable domain, variable load, variable properties.
• Necessary conditions of Weierstrass type.
• Chattering regimes and homogenization.

Homogenization of parabolic and hyperbolic equations.

• Problems
• Stochastic averaging
• Waves and dissipation

II semester

Part 3. Quasiconvexity, Bounds, G-closure

Quasiconvexity.

• Definitions
• Translation method
• Minimizing sequences
• Minimal extensions.

Bounds.

• Bounds on conducting constants
• Bounds on elastic constants
• Some other bounds

Variational problems

• Optimization of stiffness (conductivity)
• Optimization of eigenfrequencies

Various optimization problems.

• Optimization of single-loaded system by arbitrary criterium
• Min-max problems of optimization: load versus structure
• Optimization and bio-materials. What the nature wants?