Inverse optimal control with polynomial optimization

In the context of optimal control, we consider the inverse problem of Lagrangian identification given system dynamics and optimal trajectories. Many of its theoretical and practical aspects are still open. Potential applications are very broad as a reliable solution to the problem would provide a powerful modeling tool in many areas of experimental science. We … Read more

Mean squared error minimization for inverse moment problems

We consider the problem of approximating the unknown density $u\in L^2(\Omega,\lambda)$ of a measure $\mu$ on $\Omega\subset\R^n$, absolutely continuous with respect to some given reference measure $\lambda$, from the only knowledge of finitely many moments of $\mu$. Given $d\in\N$ and moments of order $d$, we provide a polynomial $p_d$ which minimizes the mean square error … Read more

Convex computation of the region of attraction of polynomial control systems

We address the long-standing problem of computing the region of attraction (ROA) of a target set (typically a neighborhood of an equilibrium point) of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints. We show that the ROA can be computed by solving a convex linear programming (LP) problem over the … Read more

Inner approximations for polynomial matrix inequalities and robust stability regions

Following a polynomial approach, many robust fixed-order controller design problems can be formulated as optimization problems whose set of feasible solutions is modelled by parametrized polynomial matrix inequalities (PMI). These feasibility sets are typically nonconvex. Given a parametrized PMI set, we provide a hierarchy of linear matrix inequality (LMI) problems whose optimal solutions generate inner … Read more

Finding largest small polygons with GloptiPoly

A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices $n$. Many instances are already solved in the literature, namely for all odd $n$, and for $n=4, 6$ and $8$. Thus, for even $n\geq 10$, instances of this … Read more

Projection methods in conic optimization

There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called … Read more

GloptiPoly 3: moments, optimization and semidefinite programming

We describe a major update of our Matlab freeware GloptiPoly for parsing generalized problems of moments and solving them numerically with semidefinite programming. Citation 28 June 2007 Article Download View GloptiPoly 3: moments, optimization and semidefinite programming

SDLS: a Matlab package for solving conic least-squares problems

This document is an introduction to the Matlab package SDLS (Semi-Definite Least-Squares) for solving least-squares problems over convex symmetric cones. The package is shortly presented through the addressed problem, a sketch of the implemented algorithm, the syntax and calling sequences, a simple numerical example and some more advanced features. The implemented method consists in solving … Read more

Nonlinear optimal control: Numerical approximations via moments and LMI-relaxations

We consider the class of nonlinear optimal control problems with all data (differential equation, state and control constraints, cost) being polynomials. We provide a simple hierarchy of LMI-relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value. Preliminary results show that good approximations are obtained with few moments. Citation LAAS … Read more

Analysis of a Belgian Chocolate Stabilization Problem

We give a detailed numerical and theoretical analysis of a stabilization problem posed by V. Blondel in 1994. Our approach illustrates the effectiveness of a new gradient sampling algorithm for finding local optimizers of nonsmooth, nonconvex optimization problems arising in control, as well as the power of nonsmooth analysis for understanding variational problems involving polynomial … Read more