Semidefinite programming for polynomial control systems 2013-04-16 10:00:00

Following original ideas of Liouville (1838), Poincaré (1899), Carleman (1932), Krylov-Bogoliubov (1937) and L. C. Young (1969), many nonconvex nonlinear infinite-dimensional optimization problems can be reformulated into convex linear programming (LP) problems in a Banach space of measures. Recent developments in functional analysis and real algebraic geometry can be exploited to solve numerically these measure LPs with the help of semidefinite programming (SDP), via a converging hierarchy of finite-dimensional LPs in the cone of positive semidefinite matrices. In this talk we apply these techniques to solve the problem of estimating the region of attraction of controlled ODEs with polynomial vector field and semialgebraic state and control constraints. We first reformulate this problem as an conic Banach LP involving the Liouville continuity (advection) PDE on occupation measures. Then we apply our hierarchy of SDP problems to generate nested semialgebraic outer approximations converging almost uniformly to the region of attraction.