International audienceThe book provides an introduction to sub-Riemannian geometry and optimal transport and presents some of the recent progress in these two fields. The text is completely self-contained: the linear discussion, containing all the proofs of the stated results, leads the reader step by step from the notion of distribution at the very beginning to the existence of optimal transport maps for Lipschitz sub-Riemannian structure. The combination of geometry presented from an analytic point of view and of optimal transport, makes the book interesting for a very large community. This set of notes grew from a series of lectures given by the author during a CIMPA school in Beirut, Lebanon

Given a locally defined, nondifferentiable but Lipschitz Lyapunov function, we construct a (discontinuous) feedback law which stabilizes the underlying system to any given tolerance. A further result shows that suitable Lyapunov functions of this type exist under mild assumptions. We also establish a robustness property of the feedback relative to measurement error commensurate with the sampling rate of the control implementation scheme.

Given a compact Riemannian manifold, we study the regularity of the optimal transport map between two probability measures with cost given by the squared Riemannian distance. Our strategy is to define a new form of the so-called Ma-Trudinger-Wang condition and to show that this condition, together with the strict convexity on the nonfocal domains, implies the continuity of the optimal transport map. Moreover, our new condition, again combined with the strict convexity of the nonfocal domains, allows us to prove that all injectivity domains are strictly convex too. These results apply, for instance, on any small C 4 -deformation of the 2-sphere.

We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier-McCann's Theorem proving existence and uniqueness of the optimal transport map. We show the absolute continuity property of Wassertein geodesics, and we address the regularity issue of the optimal map. In particular, we are able to show its approximate differentiability a.e. in the Heisenberg group (and under some weak assumptions on the measures the differentiability a.e.), which allows to write a weak form of the Monge-Ampère equation.

Abstract. We study the general problem of stabilization of globally asymptotically controllable systems. We construct discontinuous feedback laws, and particularly we make it possible to choose these continuous outside a small set (closed with measure zero) of discontinuity in the case of control systems which are affine in the control; moreover this set of singularities is shown to be repulsive for the Carathéodory solutions of the closed-loop system under an additional assumption.Key words. asymptotic controllability, control-Lyapunov function, feedback stabilization, nonsmooth analysis AMS subject classifications. 93D05, 93D20, 93B05, 34D20, 49J52, 49L25, 70K15PII. S03630129003753421. Introduction. In a previous paper [23] we considered the stabilization problem for standard control systems. In particular, we proved that if a control system is globally asymptotically controllable, then one can associate to it a control-Lyapunov function which is semiconcave outside the origin. The goal of this article is to show the utility of the semiconcavity of such functions in the construction of stabilizing feedbacks.We consider a standard control system of the general formẋ = f (x, u) which is globally asymptotically controllable, our objective being to design a feedback law u : R n → U such that the origin of the closed-loop systemẋ = f (x, u(x)) is globally asymptotically stable. Unfortunately, as pointed out by Sontag and Sussmann [28] and by Brockett [7], a continuous stabilizing feedback fails to exist in general. In addition to that, a smooth Lyapunov function may not exist either. As a matter of fact, although smooth Lyapunov-like techniques have been successfully used in many problems in control theory, it was shown by many authors (see Artstein [5] for the affine case, and more recently Clarke, Ledyaev, and Stern [11] for the general case) that there is no hope of obtaining a smooth Lyapunov function in the general case of globally asymptotically controllable systems. (The existence of such a function is indeed equivalent to that of a robust stabilizing feedback; see [17,21].) These facts lead us to consider nonsmooth control-Lyapunov functions and particularly semiconcave control-Lyapunov functions; we proved the existence of such a function in our previous article [23]. This article builds on this result to derive a useful and direct construction of stabilizing feedbacks. In fact, the semiconcavity of the control-Lyapunov function allows us to give an explicit formula for the design of the stabilizing feedbacks. More particularly, this formula can be used in the context of control systems which are affine in the control to extend Sontag's formula [26] to the case of discontinuous feedback laws. Furthermore, the main result of this paper asserts that when the control system is affine in the control, we can design a feedback which is continuous on an open dense set and which stabilizes the closed-loop system in the sense of Carathéodory solutions. Surprisingly, we show that in this case, under an additional ...

Abstract. Given a locally Lipschitz control system which is globally asymptotically controllable to the origin, we construct a control-Lyapunov function for the system which is Lipschitz on bounded sets and we deduce the existence of another one which is semiconcave (and so locally Lipschitz) outside the origin. The proof relies on value functions and nonsmooth calculus.

We prove that a Riemannian manifold (M, g), close enough to the round sphere in the C 4 topology, has uniformly convex injectivity domainsso M appears uniformly convex in any exponential chart. The proof is based on the Ma-Trudinger-Wang nonlocal curvature tensor, which originates from the regularity theory of optimal transport. Contents 14 6. Convexity of injectivity domains 24 References 32

Let (M, ∆, g) be a sub-Riemannian manifold and x0 ∈ M. Assuming that Chow's condition holds and that M endowed with the sub-Riemannian distance is complete, we prove that there exists a dense subset N1 of M such that for every point x of N1, there is a unique minimizing path steering x0 to x, this trajectory admitting a normal extremal lift. If the distribution ∆ is everywhere of corank one, we prove the existence of a subset N2 of M of full Lebesgue measure such that for every point x of N2, there exists a minimizing path steering x0 to x which admits a normal extremal lift, is nonsingular, and the point x is not conjugate to x0. In particular, the image of the sub-Riemannian exponential mapping is dense in M , and in the case of corank one is of full Lebesgue measure in M .