Stratified semiconcave control-Lyapunov functions and the stabilization problem

International audienceGiven a globally asymptotically controllable control system, we construct a control-Lyapunov function which is stratified semiconcave;that is, roughly speaking whose singular set has a Whitney stratification. Then we deduce the existence of smooth feedbacks which make the closed-loop system almost globally asymptotically stable

On continuation and convex Lyapunov functions

Given any two continuous dynamical systems on Euclidean space such that the origin is globally asymptotically stable and assume that both systems come equipped with -- possibly different -- convex smooth Lyapunov functions asserting the origin is indeed globally asymptotically stable. We show that this implies those two dynamical systems are homotopic through qualitatively equivalent dynamical systems. It turns out that relaxing the assumption on the origin to any compact convex set or relaxing the convexity assumption to geodesic convexity does not alter the conclusion. Imposing the same convexity assumptions on control Lyapunov functions leads to a Hautus-like stabilizability test. These results ought to find applications in optimal control and reinforcement learning.Comment: 16 pages, comments are welcome. V2: fixed 1 typ

Asymptotic controllability and Lyapunov-like functions determined by Lie brackets

For a given closed target we embed the dissipative relation that defines a control Lyapunov function in a more general differential inequality involving Hamiltonians built from iterated Lie brackets. The solutions of the resulting extended relation, here called degree-k control Lyapunov functions (k>= 1), turn out to be still sufficient for the system to be globally asymptotically controllable to the target. Furthermore, we work out some examples where no standard (i.e., degree-1) smooth control Lyapunov functions exist while a C^infty degree-k control Lyapunov function does exist, for some k>1. The extension is performed under very weak regularity assumptions on the system, to the point that, for instance, (set valued) Lie brackets of locally Lipschitz vector fields are considered as well

Singularities of viscosity solutions and the stabilization problem in the plane

International audienceWe study the general problem of globally asymp- totically 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 feedbacks in the plane

Nearly Optimal Patchy Feedbacks for Minimization Problems with Free Terminal Time

The paper is concerned with a general optimization problem for a nonlinear control system, in the presence of a running cost and a terminal cost, with free terminal time. We prove the existence of a patchy feedback whose trajectories are all nearly optimal solutions, with pre-assigned accuracy.Comment: 13 pages, 3 figures. in v2: Fixed few misprint