  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). 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.
