== TMA4183, 2017 ==

The a-posteriori syllabus of the course consists of chapters 1, 2 and 4 from Troltzsch, plus an overview of auxiliary results from functional analysis (chapter 7).  Below I try to summarize the list of important concepts, which I hope you can explain at the oral exam.  No proofs are required. 

* Normed, inner product, Banach and Hilbert spaces.  The dual space.  Reflexivity. Riesz representation theorem.
* Normed vs weak convergence in Banach spaces.
* (Sequential) Compactness.  Weak sequential compactness.
* Lower semi-continuity and weak lover semi-continuity.
* Existence of optimal solutions to optimization problems.
* Various derivative concepts: weak derivatives (as used in Sobolev spaces/weak formulations of PDEs); directional derivatives; Gateaux differentiability; Frechet differentiability; chain rule.  You should be in a position to compute derivatives in some simple example cases.
* Weak formulation of PDEs (at least explain on some examples).
* Existence of solutions to linear elliptic PDEs - at least in the case of symmetric bounded and coercive bilinear forms.
* The concept of control-to-state operator.  The concept of a reduced const function.
* Adjoint linear operator in Banach and Hilbert spaces.
* Concept of adjoint problem as it appears in optimal control problems.  Derivation/uses
* Derivatives of the reduced cost function.
* First order necessary optimality conditions, both in the abstract case of optimization problems over convex sets, and in the case of control problems.
* Nemytski operators: continuity, differentiability (only a brief overview)
* Semi-linear elliptic operators.  Existence of solutions using Browder-Minty theorem (at least be able to relate the idea to the monotone non-linear equations in 1D)
* Formal Lagrange method (ability to use it)
* Compact linear operators, compact embedding of function spaces, and where are they used in the context of optimal control
* Regularity of optimal controls - at least explain on an example what this is about and how we may get it from  optimality conditions
* Numerical methods:  at least the basic idea about the gradient-descent related methods (conditional/projected).


A number of these concepts are featuring in the project.  I hope you could see and point out some of these relations.

I am in the process of finding a sensor for the exam and will send a tentative schedule to you soon.  Tentatively I am aiming at 29.05 in the afternoon.