Optimization on linear matrix inequalities for polynomial systems control

These lecture notes were written for a tutorial course given during the conference "Journées Nationales de Calcul Formel" held at Centre International de Rencontres Mathématiques, Luminy, Marseille, France in May 2013. They are aimed at giving an elementary and introductory account to recent applications of semidefinite programming to the numerical solution of decision problems involving polynomials in systems and control theory. The main technical results are gathered in a hopefully concise, notationally simple way, but for the sake of conciseness and readability, they are not proved in the document. The reader interested in mathematical rigorous comprehensive technical proofs is referred to the papers and books listed in the "Notes and references" section of each chapter. Comments, feedback, suggestions for improvement of these lectures notes are much welcome.National audienceMany 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). For some of these decision and optimization problems, it is hoped that the numerical solutions computed by SDP can be refined a posteriori and certified rigorously with appropriate techniques