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Institut Girard Desargues , , SINGULARITIES OF SOME VISCOSITY SUPERSOLUTIONS AND THE STABILIZATION PROBLEM IN THE PLANE LUDOVIC RIFFORD Abstract. We study the general problem of globally asymptotically controllable affine systems in the plane. As preliminaries we present some results of independent interest. We study the regularity of some sets related to semiconcave viscosity supersolutions of Hamilton-Jacobi- Bellman equations. Then we deduce a construction of stabilizing feed- backs in the plane. 1. Introduction This paper is concerned with the stabilization problem for a control sys- tem of the form x˙ = f(x,?) = f0(x) + m ∑ i=1 ?ifi(x), ? = (?1, · · · ,?m) ? Bm, (1.1) assuming that the maps f0, f1, · · · , fm are locally Lipschitz from R2 into R2. It is well known that, even if every initial state can be steered to the origin by an open-loop control ?(t) (t ? [0,∞)), there may not exist a continuous feedback control ? = ?(x) which locally stabilizes the system (1.1). The purpose of this paper is to build upon the work which was initiated by the author in [16] and [15] in order to provide specific results in the case of the stabilization problem in the plane.