2013
DOI: 10.1007/s00526-013-0627-3
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The Hopf–Lax formula in Carnot groups: a control theoretic approach

Abstract: The purpose of this paper is to bring a new light on the state-dependent HamiltonJacobi 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 formwhere X 1 , . . . , X m are a basis of the first layer of the Lie algebra of the group G, and Ψ : R m → 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 u(x, t) = infwhere… Show more

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Cited by 12 publications
(25 citation statements)
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“…Following the same arguments of [23, Theorem 2, Section 10.3.3] and [8], we get that u ε fulfills u ε (0, x) = g(x) and it is a viscosity solution of…”
Section: Proof Of Theorem 31mentioning
confidence: 70%
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“…Following the same arguments of [23, Theorem 2, Section 10.3.3] and [8], we get that u ε fulfills u ε (0, x) = g(x) and it is a viscosity solution of…”
Section: Proof Of Theorem 31mentioning
confidence: 70%
“…We only sketch the proof; for the detailed calculations we refer the reader to [23, Section 10.3.3] and to [8]. First of all we prove that u ε is a solution of (3.1).…”
Section: Proof Of Theorem 31mentioning
confidence: 91%
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“…We refer the reader to Theorem 4.4, which answers a question asked in [26]. A more general question on Lipschitz continuity of viscosity solutions was posed in [2], but it is not clear if our method here immediately applies to that general setting.…”
Section: )mentioning
confidence: 99%