Course on polynomial optimization and optimal control
(Didier Henrion, LAAS-CNRS, Toulouse, France and FEL-CVUT, Prague, Czech Republic)

Czech Technical University, Prague, Czech Republic - 19-21 February 2018

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). It consists of six two-hour lectures, given on Monday 19, Tuesday 20 and Wednesday 21 February 2018, from 10am to noon and from 2pm to 4pm.

The course is given in room E-24, ground floor on your left hand side 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.


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.


This is a course for graduate students or researchers with a background in linear control systems, linear algebra and convex optimization.

First, we survey a mathematical technology introduced in 2000 by Jean Bernard Lasserre to solve globally non-convex optimization problems on multivariate polynomials with the help of a hierarchy of convex semidefinite programming problems (linear matrix inequalities or LMI = linear programming problems in the cone of positive semidefinite matrices). Instrumental to the development of this technique is the duality between the cone of positive polynomials (real algebraic geometry) and the cone of moments (functional analysis). These basic objects are introduced and studied in detail, with a special focus on conic optimization duality, and some illustrative examples are described.

Then in a second part, we study polynomial optimal control, which consists of minimizing a polynomial Lagrangian over a polynomial vector field subject to semi-algebraic control and state constraints, typically a nonconvex problem for which there is no solution in classical Lebesgue spaces. To overcome this, polynomial optimal control problems are first formulated as linear programming (LP) problems in the cone of occupation measures (standard objects in Markov decision processes and ergodic theory of dynamical systems), and infinite-dimensional convex duality is used to establish the link with subsolutions of the Hamilton-Jacobi-Bellman (HJB) partial differential equation satisfied by the value function. Then, the Lasserre hierarchy is applied to solve numerically these infinite-dimensional LP problems.

In the third part, we address the problem of computing the region of attraction (ROA) of a target set (e.g., a neighborhood of an equilibrium point) of a polynomial control system. We show that the ROA can be computed by solving an infinite-dimensional LP over occupation measures. In turn, this problem can be solved approximately with the Lasserre hierarchy. Our approach is genuinely primal in the sense that convexity of the problem of computing the ROA is an outcome of optimizing directly over system trajectories. The dual infinite-dimensional LP on nonnegative continuous functions (approximated by polynomial sum-of-squares) allows us to generate a hierarchy of semialgebraic outer approximations of the ROA at the price of solving a sequence of LMI problems with asymptotically vanishing conservatism.

In the final part, we develop further our LP approach for the optimal control of nonlinear switched systems where the control is the switching sequence. This is done by introducing modal occupation measures, which allow to relax the problem as a primal LP. Its dual LP of HJB inequalities is also characterized. The LPs are then solved numerically with the Lasserre hierarchy. Because of the special structure of switched systems, we obtain a much more efficient method than could be achieved by applying the standard Lasserre hierarchy for general optimal control problems.

Outline and lecture notes

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.

    The material on polynomial optimal control and HJB inequalities can be found in this document:

  • D. Henrion, E. Pauwels. Linear conic optimization for nonlinear optimal control. Chapter 10 on pages 121-134 in S. Ahmed, M. Anjos, T. Terlaky (Editors). Advances and Trends in Optimization with Engineering Applications. MOS-SIAM series, SIAM, Philadelphia, 2017.

    The material on approximating the region of attraction of polynomial control systems can be found in this document:

  • D. Henrion, M. Korda. Convex computation of the region of attraction of polynomial control systems. IEEE Transactions on Automatic Control 59(2):297-312, 2014

    as well as in the following Matlab codes:

  • ROA - Matlab codes (including distributions of YALMIP and GloptiPoly 3, and using either MOSEK or SeDuMi which should be installed) for computing estimates of the region of attraction of a polynomial control system. Developed by Milan Korda.

    The material on optimal switching control can be found in this document:

  • M. Claeys, J. Daafouz, D. Henrion. Modal occupation measures and LMI relaxations for nonlinear switched systems control, Automatica, 64(2):143-154, 2016

    as well as in the following Matlab codes:

  • Switch - Matlab codes for optimal control of nonlinear switching systems. Developed by Mathieu Claeys.

    Homeworks and exam

    Homeworks are given during the course. A written examination can be organized. Certificates of attendance can be provided on request.

    Last updated on 1 February 2018.