### MATH 2603 Fall Semester 2022

### Advanced Scientific Computing - Optimal Control of Partial Differential Equations

MWF 1:00-1:50PM,
Thackeray Hall 524

### Office Hours

###
MW 2:00 - 3:00, and by appointment

Office: Thackeray 612

E-mail: trenchea@pitt.edu

This course is an introduction to modern for optimal control and optimization.
Optimal control theory is concerned with finding control functions that minimize cost functions for systems described by differential equations.
The methods have found widespread applications in aeronautics, mechanical engineering, the life sciences and other disciplines.
We shall focus on optimal control problems where the state equation is an elliptic of parabolic partial differential equation.
Included are topics such as the existence of optimal solutions, necessary optimality conditions and adjoint equations,
second-order sufficient conditions, numerical algorithms and implementation techniques.
The expositions begins with control problems with linear equations, quadratic cost functions and control constraints.
The
course
is self-contained, basic facts on weak solutions of elliptic and parabolic equations, principles of
functional analysis are introduced and explained as they are needed.
Many simple examples illustrate the theory and its hidden difficulties.

Written homework and several computational projects will be assigned.
The suggested programming language for computer assignments is Matlab, software produced
by The MathWorks.
The Matlab language provides extensive library of
mathematical and
scientific function calls entirely built-in.
Matlab
is available on Unix and Windows in the university computing labs.
The full set of manuals is on the web in
html
and also in
Adobe PDF format. The "Getting Started" manual is a good
place to begin and is available both in html
format and in Adobe
PDF format. The full reference manual as well as manuals for each
of the many toolboxes are all available.

### Course materials

Lecture notes and research papers from published literature.
*
Optimal Control Applied to Biological Models* (Chapman & Hall / Crc Mathematical and Computational Biology), by Suzanne Lenhart and John T. Workman, 2007.

The
*MATLAB m-files* needed for the labs.
Other computer * Matlab exercises* for optimal control,
by Dr. John Burkardt.
* Convexity and Optimization in Banach Spaces*, Springer, by Viorel Barbu and Theodor Precupanu, 2012.
* Optimal Control from Theory to Computer Programs*, Springer, by Viorel Arnautu, and Pekka Neittaanmaki, 2003.
*An Introduction to Applied Optimal Control*, Academic Press, by Greg Knowles, 1981.

**Additional references**
Optimal Control of Distributed Systems. Theory and Applications, by
A. V. Fursikov.
Optimal Control Theory of Partial Differential Equations. Theory, Methods and Applications, by
Fredi Tröltzsch.
Optimization with PDE constraints, by
Michael Hinze,
Rene Pinnau, Michael Ulbrich and Stefan Ulbrich.
An Introduction to Optimal Control Problems in Life Sciences and Economics
From Mathematical Models to Numerical Simulation with MATLAB, by
Sebastian Anita,
Viorel Arnautu,
Vincenzo Capasso.
*Perspectives in Flow Control and Optimization*, SIAM Advances in Design and Control, by Max D. Gunzburger, 2002.
*An Introduction to Mathematical Optimal Control Theory*, by Lawrence C. Evans.
*Lagrange Multiplier Approach to Variational Problems and Applications*, SIAM Advances in Design and Control 15, by K. Ito and K. Kunisch, 2008.
* Representation and Control of Infinite Dimensional Systems*, Birkhauser, 2nd edition, by Alain Bensoussan, Giuseppe Da Prato, Michel Delfour, Sanjoy K. Mitter, 2007.
* Optimal control of variational inequalities*, Pitman Advanced Publishing Program, by Viorel Barbu, 1984.
* Analysis and Control of Nonlinear Infinite Dimensional Systems*, Academic Press, by Viorel Barbu, 1993.
* Some Aspects of the Optimal Control of Distributed Parameter Systems*, SIAM Philadelphia, by J.L. Lions.
* Numerical Methods for Variational Inequalities and Optimal Control Problems*, University "Al.I. Cuza" Iasi, Romania, by Viorel Arnautu, 1997.

#### Disability Resource Services

If you have a disability for which you are or may be requesting an
accommodation, you are encouraged to contact both your instructor and
Disability Resources and Services, 140 William Pitt Union,
412-648-7890 or 412-383-7355 (TTY) as early as possible in the term. DRS will verify your disability and determine reasonable accommodations for this course.
#### Academic Integrity

Cheating/plagiarism will not be tolerated.
Students suspected of violating the University of Pittsburgh Policy on
Academic Integrity will incur a minimum sanction of a zero score for the quiz,
exam or paper in question. Additional sanctions may be imposed,
depending on the severity of the infraction.
On homework, you may work with other students or use library resources,
but each student must write up his or her solutions independently.
Copying solutions from other students will be considered cheating, and handled
accordingly.
#### Statement on Classroom Recording

To address the issue of students recording a lecture or class session, the University's
Senate Educational Policy Committee issued the recommended statement on May 4, 2010.
"To ensure the free and open discussion of ideas, students may not record classroom lectures,
discussion and/or activities without the advance written permission of the instructor,
and any such recording properly approved in advance can be used solely for the student's own private use."