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.
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Part 2. Homogenization and control in PDE.
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
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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
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Part 4. Structural Optimization .
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?
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