6533b873fe1ef96bd12d4a30

RESEARCH PRODUCT

Etude asymptotique et transcendance de la fonctionvaleur en contrôle optimal. Catégorie log-exp en géométrie sous-Riemannienne dans le cas Martinet.

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The main subject of this work is the study and the role ofabnormal trajectories in optimal control theory.We first recall some fundamental results in optimal control. Thenwe investigate the optimality of abnormal trajectories forsingle-input affine systems with constraint on the control, firstfor the time-optimal problem, and then for any cost, the finaltime being fixed or not.Using such an affine system,we extend this theory to sub-Riemannian systems of rank 2.These results show that, under general conditions, an abnormaltrajectory is \it{isolated} among all solutions of the systemhaving the same limit conditions, and thus is \it{locallyoptimal}, until a first \it{conjugate point} which can becharacterized.Then we investigate the asymptotic behaviour and the regularityof the value function associated to an analytic affine systemwith a quadratic cost. We prove that, if there is no abnormalminimizer, then the value function is \it{subanalytic andcontinuous}. If there exists an abnormal minimizer, thesubanalytic category is not large enough in general, notably insub-Riemannian geometry. The existence of an abnormal minimizeris responsible for \it{non-properness} of the exponentialmapping, which implies a phenomenon of \it{tangency} of the levelsets of the value function with respect to the abnormaldirection. In the single-input affine case, or in thesub-Riemannian case of rank 2, we describe precisely thiscontact, and we get a partition of the sub-Riemannian sphere nearthe abnormal into two sectors called \it{$L^\infty$-sector} and\it{$L^2$-sector}.\\The question of transcendence is studied in the Martinetsub-Riemannian case where the distribution is$\Delta=\rm{Ker }(dz-\f{y^2}{2}dx)$. We prove that for a generalgradated metric of order $0$~:$g=(1+\alpha y)^2dx^2+(1+\beta x+\gamma y)^2dy^2$,spheres with small radii \it{are not subanalytic}. In the generalintegrable case where $g=a(y)dx^2+c(y)dy^2$, with $a$ and $c$analytic, Martinet spheres belong to the \it{log-exp category}.