##
Course on polynomial and LMI optimization with applications in control

(Didier Henrion, LAAS-CNRS, Toulouse, France and FEL-CVUT, Prague,
Czech Republic)

### Czech Technical University,
Prague, Czech Republic - 16-23 February 2015

## Venue and dates

The course is given at the Charles Square campus of the
Czech Technical University, in the historical center of
Prague (Karlovo Namesti 13, 12135
Praha 2) during the second half of February 2015.
It consists of six two-hour lectures, given on Monday 16,
Thursday 19 and Monday 23 February, from 10am to noon
and from 2pm to 4pm.
The course is given in room E24,
ground floor, to your left handside when entering building E.
Please refer to these maps
for instructions to reach this building. The campus entrance
is restricted to badge holders, so please ask the doorperson
to let you in through the electronic gates.

## Registration

The course is primarily aimed at students from the Czech Technical
University in Prague, yet external participants are welcome. There is
no registration fee. Please note that the Czech Technical University
will not provide assistance regarding traveling and accomodation in
Prague.

## Description

This is a course for graduate students or researchers
with a background in linear control
systems, linear algebra and convex optimization.
Many problems of systems control theory boil down to solving polynomial equations,
polynomial inequalities or polyomial differential equations. Recent advances in convex
optimization and real algebraic geometry can be combined to generate approximate
solutions in floating point arithmetic.

In the first part of the course we describe semidefinite programming (SDP) as an extension
of linear programming (LP) to the cone of positive semidefinite matrices. We investigate
the geometry of spectrahedra, convex sets defined by linear matrix inequalities (LMIs)
or affine sections of the SDP cone. We also introduce spectrahedral shadows, or lifted
LMIs, obtained by projecting affine sections of the SDP cones. Then we review existing
numerical algorithms for solving SDP problems.

In the second part of the course we describe several recent applications of SDP. First, we
explain how to solve polynomial optimization problems, where a real multivariate polynomial
must be optimized over a (possibly nonconvex) basic semialgebraic set. Second, we
extend these techniques to ordinary differential equations (ODEs) with polynomial
dynamics, and the problem of trajectory optimization (analysis of stability or performance
of solutions of ODEs). Third, we conclude this part with applications to optimal control
(design of a trajectory optimal w.r.t. a given functional).

## Outline

The course closely follows the lecture notes:

D. Henrion. Optimization on linear matrix inequalities
for polynomial systems control. International Summer School of Automatic Control, Grenoble, France, September 2014.

## Homeworks and exam

Homeworks are given during the course.
A written examination can be organized.

Last updated on 5 November 2014.