# [TMA4183 2016v] Optimization II: Optimal Control of PDEs

## Messages

• 03.05: I have now received the project reports from most of you. I will read through them over the weekend/beginning of the next week. Regarding the oral exam: I am available in the period 23.05-26.05, 01.06-03.06, 07.06-09.06. Please email me and indicate if there are any preferred dates/time periods (morning/afternoon) when you are available. We will use your project reports as a starting point for the discussion at the exam.
• 25.04: I will do another "overview" lecture: large scale problems, regularization.
• 18.04: Let us take an "overview" type lecture tomorrow (19.04). I can say a few words on control of time-dependent (say, parabolic) PDEs and some use of control theory in finite element method (least squares Petrov-Galerkin methods with discontinuous test functions)
• 18.04: Dear everyone, it seems that there some of you who are still working on the project. Please pop in today 12-16 to 1038/SBII if you have any questions.
• 14.04: Dear everyone, could you please write me a short project status email? I just want to know where you are, whether you need more time, and if so how much.
• 11.04: In the project you can assume that $\Omega$ is of class $C^{1,1}$ or convex, not just Lipschitz. This way the results about the elliptic regularity on p. 113 in [Tr] apply. Note that the continuity of the state can be inferred from the Sobolev embedding theorem (see Theorem 7.1 in [Tr]).
• 04.04: I have finally published the course project. There will be no lectures in the period 04.04-15.04 to let you focus on the project. I am available for questions during this period. Please let me know if you need more time, I am quite flexible with the last two weeks of the course.
• 18.01: The schedule has changed! According to the doodle the best time slot for the lectures is Tue 14:15-16:00, and for the exercises is probably Thu 10:15-11:00. I have reserved 656, SBII for both Tue and Thu for the remaining weeks of the semester (One exception: week 10, Thu is in room 822, SBII).
• 13.01: Note that owing to a collision with another course I cannot make it to the exercise session. My suggestion is for you to try to work on the exercises on your own. Email me with any questions.
• 12.01: I understand that there is a clash between the lecture on Wed and Experts in Teams. Additionally, I have a clash for the planned exercise session on Thu 12-14 and another course that I teach this semester. Let us first focus on trying to reschedule the lecture. Could you please fill out the this doodle and indicate time-slots (2 hours), which would suite you on a typical week this semester? That is, ignore the dates, only concentrate on the week days/times. I would really prefer avoiding Thu and 08:15 slots, if possible.
• 02.01: Happy 2016 and welcome to the course!

## Synopsis

Linear and non-linear partial differential equations (PDEs) constitute one of the most widely used mathematical framework for modelling various physical or technological processes, such as fluid flow, structural deformations, propagation of acoustic and electromagnetic waves among countless other examples. Improvement in such processes therefore require modelling and solving optimization problems constrained with PDEs, and more generally convex and non-convex optimization problems in spaces of functions.

In this course you will learn the theory pertinent for analysing optimization problems of this type and also fundamental numerical methods for solving these problems. We will mostly concentrate on the optimal control of processes governed with linear and semilinear elliptic PDEs.

We will aim at a reasonably self-contained course (and the textbook supports this, too) but of course some knowledge of PDEs, functional analysis, and optimization theory is beneficial.

## Learning objectives

After meeting the learning objectives of the course, the student will be able to:

• analyze control-to-state operators for model control problems
• derive necessary and sufficient optimality conditions for optimal control problems with or without state constraints
• assess existence of solutions to model optimal control problems
• implement optimization algorithms on a computer
• apply optimization algorithms to model problems
• explain the basic properties of the relevant functional spaces, in particular Sobolev spaces

The textbook is Tröltzsch.

• F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods, and Applications, AMS Graduate Studies in Mathematics, v. 112, 2010. Amazon AMS
• J.C. De los Reyes, Numerical PDE-Constrained Optimization, 2015. Available online via Springer link
• M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich, Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, v. 23, Springer, 2009. Available online via Springer link
• Eds.: Lorenz T. Biegler, M. Heinkenschloss, O. Ghattas, B. van Bloemen Waanders, Large-Scale PDE-Constrained Optimization, 2003. Available online via Springer link
• Eds.: L.T. Biegler , O. Ghattas , M. Heinkenschloss , D. Keyes and B. van Bloemen Waanders, Real-Time PDE-Constrained Optimization, 2007. Available online via SIAM
• L.C. Evans, Partial Differential Equations, AMS Graduate Studies in Mathematics, v. 19, 2010. Amazon AMS

## Schedule

<schedule> <timeslots>08:15-09:00|09:15-10:00|10:15-11:00|11:15-12:00|12:15-13:00|13:15-14:00|14:15-15:00|15:15-16:00|16:15-17:00</timeslots> <entry day:Mo start:4 end:5 color:ntnu2>Lecture (KJL22)</entry> <entry day:Tu start:6 end:7 color:ntnu2>Lecture (656/SBII)</entry> <entry day:Th start:2 end:2 color:ntnu2>Exercises (656/SBII; week 10: 822/SBII)</entry> </schedule>

## Lecture plan and log

[Tr]=Tröltzsch

Date Topic Reference
19.04 Overview of the structure & computational challenges with optimal control of time-dependent PDEs. Petrov-Galerkin methods with discontinuous test functions (DPG) and their relation to optimal control Chapter 3 [Tr], this paper on DPG, a couple of FENICS implementations of DPG for Laplace equation poisson_dpg_primal.py, poisson_dpg.py. Both scripts simply produce files, visualize them with Paraview
15.03 Intro to second order derivatives & second order sufficient optimality conditions 4.9, 4.10.1-4.10.2 [Tr]
14.03 First order necessary optimality conditions 4.5, 4.6 (skim 4.7) [Tr]
08.03 Existence of optimal controls 4.4 [Tr]
07.03 Nemytskii operators 4.3 [Tr]
01.03 Continuity of solutions. Weakening of assumptions on coercivity 4.2.3-4.2.4 [Tr]
29.02 Existence/uniqueness of solutions to semi-linear elliptic BVPs 4.1-4.2.2 [Tr]
23.02 Work on numerical exercises Section 2.12 [Tr]
22.02 Higher regularity Sections 2.14, 2.15 [Tr]
16.02 Work on numerical exercises Section 2.12 [Tr]
15.02 Work on numerical exercises Section 2.12 [Tr]
09.02 Numerical methods: Projected gradient method, Active set methods Section 2.12 [Tr]
08.02 Discretize or optimize first? Numerical methods: Conditional gradient method Section 2.12 [Tr]
02.02 Special cases of first order necessary conditions. The formal Lagrangian method. Sections 2.8, 2.10 [Tr]
01.02 First order necessary optimality conditions Sections 2.8 [Tr]
26.01 Differentiability in Banach spaces, adjoint operators Sections 2.6-2.7 [Tr]
25.01 Existence of optimal controls Section 2.5 [Tr]
19.01 Weak solutions to elliptic PDEs Section 2.3 [Tr]
18.01 Linear maps, weak convergence Section 2.4 [Tr]
13.01 Weak derivatives and Sobolev spaces Sections 2.1-2.2 [Tr]
11.01 Introduction, examples, concepts in finite dimensions Chapter 1 [Tr]

## Exercises

Exercises are not compulsory.

• Exercise 1 with hints is here. Solutions. Please read Chapter 1 and Sections 2.1-2.2 in [Tr].
• Exercise 2 with hints is here. Solutions. Please review Sections 2.1-2.4 in [Tr].
• Exercise 3 with hints is here. Solutions. Please review Sections 2.5-2.7 in [Tr].
• Exercise 4 is here. Solutions. Please review Sections 2.8-2.10 in [Tr]. Here is a suggestion about a first numerical exercise (Apparently an even easier test-case is provided by $y(x,y)=\sin(2\pi x)\sin(2\pi y)$, whose Laplacian is also 0 on the boundary.) It is very open and we should perhaps agree upon how we proceed with exercises like this in the future. I therefore highly recommend that you try solving it or at least looking at the provided FEniCS-based solution. fvmlaplace.pdf fvmlaplace.zip ex4_num.py
• Here is an exercise, in which you are asked to implement and test three different algorithms described in Chapter 2 of [Tr]. There will be no lectures during week 7 (15.02-19.02); please catch up with the exercises.
• Exercise 6 (proof that projection onto box constraints does not ruin the weak differentiability) with hints is here. Solutions.

## Project

The course project description is here. The problem structure is such that you should be able to update and adapt the numerical code from the exercise 5.

## Exam information

The exam will test a selection of the Learning objectives. The final grade is based on the course project (10%) and oral exam (90%). The reserved date for the oral exam is 30.05.2016, from 09:00. We will discuss the exam schedule during the semester.

## Contact

• Lectures & exercises: Anton Evgrafov antone [at] math [dot] ntnu [dot] no; office hours: Mon 14:15-15:00
• Reference group: everyone (small course)
• Please put TMA4183 in the subject line when emailing