Regularity of the Minimum Time and of Viscosity Solutions of Degenerate Eikonal Equations via Generalized Lie Brackets

Abstract

In this paper we relax the current regularity theory for the eikonal equation by using the recent theory of set-valued iterated Lie brackets. We give sufficient conditions for small time local attainability of general, symmetric, nonlinear systems, which have as a consequence the Hölder regularity of the minimum time function in optimal control. We then apply such result to prove Hölder continuity of solutions of the Dirichlet boundary value problem for the eikonal equation with low regularity of the coefficients. We also prove that the sufficient condition for the Hölder regularity are essentially necessary, at least for smooth vector fields and target.

A degenerate elliptic equation for second order controllability of nonlinear systems

“…Therefore STLA is a way we approach the regularity of the minimum time function, which is the prototype of solutions of Dirichlet boundary value problems for the Hamilton Jacobi equation. For the connection, see for instance the recent paper by Bardi, Feleqi and the author [5]. It is then clear that what we derive here has precise consequences on that problem as well.…”

“…For attainability of a target different from a point we recall the papers by Bacciotti [1] in the case of targets of codimension 1 and the author [19] for manifolds of any dimension, possibly with a boundary. We mention Bardi-Falcone [3] who showed that Petrov condition is also necessary for local Lipschitz continuity of the minimum time function, while in [5] we derive necessary second order conditions. More recently the work by Krastanov and Quincampoix [9,10] pointed out the importance of the geometry of the target and studied higher order attainability of nonsmooth targets for affine systems with nontrivial drift.…”

“…For completeness, we conclude this section recalling a necessary condition which is proved in [5] for second order conditions. Recall first that when the target is the closure on an open set, it was proved in [3] that the Lipschitz continuity of the minimum time function is characterised by the existence of vector fields pointing inward the target.…”

“…If instead su = 1, s = 2/3, then we reach (1.5) with the exponent 1/3. We racall that estimates of the form(3.15) are crucial to derive local Hölder continuity of the minimum time function and for solutions to boundary value problems for the Hamilton-Jacobi equation, see[2,5].…”

HJ inequalities involving Lie brackets and feedback stabilizability with cost regulation

“…Here, the dissipative rate γ is an increasing function taking values in ]0, +∞[ and ∂ P U (x) denotes the proximal subdifferential of U at x (see (6)). We call relation (5) the degree-k HJ dissipative inequality (where HJ stands for Hamilton-Jacobi).…”