No abstract

The value function associated with an optimal control problem subject to the Navier-Stokes equations in dimension two is analyzed. Its smoothness is established around a steady state, moreover, its derivatives are shown to satisfy a Riccati equation at the order two and generalized Lyapunov equations at the higher orders. An approximation of the optimal feedback law is then derived from the Taylor expansion of the value function. A convergence rate for the resulting controls and closed-loop systems is demonstrated.

sn this reportD we stte nd prove (rstE nd seondEorder neessry onditions in ontrygin form for optiml ontrol prolems with pure stte nd mixed ontrolEstte onstrintsF e sy tht vgrnge multiplier of n optiml ontrol prolem is ontrygin multiplier if it is suh tht ontrygin9s minimum priniple holdsD nd we ll optimlity onditions in ontrygin form those whih only involve ontrygin multipliersF yur onditions rely on tehnique of prtil relxtionD nd pply to ontrygin lol minimF Key-words: yptiml ontrolY pure stte nd mixed ontrolEstte onstrintsY ontrygin9s prinipleY ontrygin multipliersY seondEorder neessry onditionsY prtil relxtionFThe research leading to these results has received funding from the EU 7th Framework Programme (FP7-PEOPLE-2010-ITN), under GA number 264735-SADCO, and from the Gaspard Monge Program for Optimization and operations research (PGMO). * Inria-Saclay and CMAP, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France.Emails: frederic.bonnans@inria.fr, xavier.dupuis@cmap.polytechnique.fr, laurent.pfeier@polytechnique.edu. Conditions nécessaires du second ordre sous formePontryaguine pour des problèmes de commande optimale Résumé : hns e rpportD nous énonçons et prouvons des onditions néessires du premier et seond ordre sous forme ontryguine pour des prolèmes de ommnde optimle ve onE trintes pures sur l9étt et mixtes sur l9étt et l ommndeF xous ppelons multipliteur de ontryguine tout multipliteur de vgrnge pour lequel le prinipe de ontryguine est stisE fit et prlons de onditions d9optimlité sous forme ontryguine si elles ne font intervenir que des multipliteurs de ontryguineF xos onditions s9ppuient sur une tehnique de relxtion prtielle et sont vlles pour des minim de ontryguineF

Using a projection-based decoupling of the Fokker-Planck equation, control strategies that allow to speed up the convergence to the stationary distribution are investigated. By means of an operator theoretic framework for a bilinear control system, two different feedback control laws are proposed. Projected Riccati and Lyapunov equations are derived and properties of the associated solutions are given. The well-posedness of the closed loop systems is shown and local and global stabilization results, respectively, are obtained. An essential tool in the construction of the controls is the choice of appropriate control shape functions. Results for a two dimensional double well potential illustrate the theoretical findings in a numerical setup.Mathematics Subject Classification. 35Q35, 49J20, 93D05, 93D15.