The Hopf–Lax formula in Carnot groups: a control theoretic approach

The purpose of this paper is to bring a new light on the state-dependent Hamilton-Jacobi equation and its connection with the Hopf-Lax formula in the framework of a Carnot group (G, ○). The equation we shall consider is of the form (Formula presented.) where X1,...,Xm are a basis of the first layer of the Lie algebra of the group G and Ψ: ℝm → ℝ is a superlinear, convex function. The main result shows that the unique viscosity solution of the Hamilton-Jacobi equation can be given by the Hopf-Lax formula (Formula presented.) where ΨG: G → ℝ is the G-Legendre-Fenchel transform of Ψ defined by a control theoretical approach. We recover, as special cases, some known results like the classical Hopf-Lax formula in the Euclidean spaces ℝn showing that Ψℝn is the Legendre-Fenchel transform Ψ* of Ψ; moreover, we recover the result by Manfredi and Stroffolini in the Heisenberg group for particular Hamiltonian function Ψ. In this paper we follow an optimal control problem approach and we obtain several properties for the value functions u and ΨG © 2013 Springer-Verlag Berlin Heidelberg.