Course contents
The first part of the course will be taught by Pavel Krejčí. The purpose of this part is to explain the motivation for modeling evolution processes in continuous media by nonlinear partial differential equations and systems. Particular attention will be paid to understanding the link between physical principles and objects, and the corresponding mathematical concepts, including for example elasticity, plasticity, hysteresis, thermodynamics, phase transitions, or shocks. Questions of existence and qualitative properties, such as uniqueness, boundedness, periodicity, or asymptotic behavior of solutions to initial-boundary-value problems for evolutionary PDE systems will be discussed.
Some of the following items (to be agreed with students) will be dealt with:
1. Linear and nonlinear diffusion, physical meaning of Dirichlet, Neumann and Robin boundary conditions, maximum principle;
2. An introduction to Sobolev spaces, embeddings, interpolations, anisotropic interpolation inequalities;
3. Methods of solving nonlinear and degenerate diffusion problems, Moser iterations;
4. Diffusion in media with hysteresis;
5. Long time behavior of solutions, stability;
6. Coupled systems in thermomechanics I: phase transitions with volume changes, thermoviscoelasticity, phase separation;
7. Coupled systems in thermomechanics II: diffusion in deformable porous media;
8. Elasticity and elastoplasticity, oscillations in elastoplastic solids, wave propagation, lower-dimensional elastoplastic structures;
9. Nonlinear hyperbolic equations, method of characteristics, shocks;
10. Riemann problem, construction of a solution, uniqueness criteria, entropy conditions;
11. Wave propagation in media with hysteresis, regularity, long time behavior of solutions;
12. Models of multifunctional materials: piezoelectricity, magnetostriction.
The second part of the course will be taught by Jürgen Sprekels. The general topic of these lectures is the optimal control of parabolic PDEs, with a clear focus on the optimal distributed and boundary control of the heat equation as the main example. While initially bringing the linear theory, it is intended to discuss semilinear problems and, if time permits, also systems of Cahn--Hilliard type later.
The following topics will be addressed:
1. Some functional analytic foundations: minimization in reflexive spaces, the role of the notions of convexity and weak sequential lower semicontinuity, directional and Fréchet differentiability, first-order necessary minimality conditions and their connection to variational inequalities.
2. Optimal distributed and/or boundary control for the linear heat equation under box constraints for the controls: existence and uniqueness of optimal controls,
derivation of necessary first-order conditions in terms of a variational inequality and the adjoint state, the formal Lagrange technique to derive the adjoint problem,
comments on the numerical solution of the optimal control problem.
3. Optimal distributed control for semilinear heat equations under box constraints for the controls: existence of optimal controls, derivation of necessary first-order conditions
(unique solvability of the linearized system, differentiability of the control-to-state operator, the formal Lagrange technique, solvability of the adjoint system).
If time permits, then also systems of Cahn--Hilliard type will be addressed.
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