### Math 2603: Advanced Scientific Computing III

*An introduction to optimal control, parameter estimation and uncertainty quantification*

**Instructor:** Catalin Trenchea

Phone: (412) 624-5681

E-mail: trenchea@pitt.edu

Office: Thackeray 606

**Lecture:** MWF 1:00-1:50 THACKERAY 525

**Office Hours:**
Tu 10:00am-noon, Thu 10:00am-noon and by appointment.

**Textbook:**
The course will not follow closely a specific text book, and
will be self-contained as far as possible.

**Prerequisites:**
Good undergraduate background in linear algebra and advanced
calculus. Familiarity with partial differential equations will be useful.

Lectures will adapt to diverse backgrounds. Please contact the instructor if you have questions about your preparation.

**Content:**
This course will focus on the study of *optimal control* and *parameter estimation* problems for distributed (deterministic and/or random) parameter systems, i.e., for systems described by a boundary value problem for partial differential equations. We will give an introduction to the development and analysis of several approaches for solving deterministic and *stochastic* optimization problems (optimal control and parameter identification), and discuss some of the many issues that arise in the practical implementation of algorithms. We will also briefly address numerical methods for *uncertainty quantification* and the analysis of stochastic partial differential equations.

**Topics to be covered:**
examples of control systems, controllability and observability, linear-time optimal control, the Pontryagin maximum principle, optimal control of systems governed by partial differential equations (elliptic control problems, necessary and sufficient conditions for optimality, boundary control, control of systems governed by parabolic equations, time optimal control), brief introduction into analysis of stochastic partial differential equations and numerical methods for uncertainty quantification.

**Homework:**
Written homework and several computational projects will be assigned.

**Exams:**
There will be one in-class midterm exam, a paper presentation, and a final
project.

**Material:**
Lecture notes and research papers from published literature.

**Other references:**

*Perspectives in Flow Control and Optimization*, SIAM Advances in Design and Control, by Max D. Gunzburger, 2002.

*An Introduction to Applied Optimal Control*, Academic Press, by Greg Knowles, 1981.

*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.

* Optimal Control Applied to Biological Models* (Chapman & Hall / Crc Mathematical and Computational Biology), by Suzanne Lenhart and John T. Workman, 2007.

* 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

* Convexity and Optimization in Banach Spaces, D. Reidel Publilshing Company*, by Viorel Barbu and Theodor Precupanu, 1985.

* Numerical Methods for Variational Inequalities and Optimal Control Problems*, University "Al.I. Cuza" Iasi, Romania, by Viorel Arnautu, 1997.

The course web page will be updated continuously throughout the semester.
The student is responsible for checking this web page for assignments
and policies.

If you have a disability for which you are or may be requesting an
accodomation, you are encouraged to contact both your instructor
and the Office of Disability Resources and Services, 216 William Pitt
Union, (412) 648-7890, as early as possible in the term. DRS will verify
your disability and determine reasonable accomodations for this course.