A comparison principle for Hamilton-Jacobi equations with discontinuous Hamiltonians

Giga, Yoshikazu; Górka, Przemysław; Rybka, Piotr

doi:10.1090/s0002-9939-2010-10630-5

Cited by 19 publications

(12 citation statements)

References 9 publications

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“…Problems with a discontinuous running cost were addressed by either Garavello and Soravia [18,19], or Camilli and Siconolfi [12] (even in an L ∞ -framework) and Soravia [27]. To the best of our knowledge, all the uniqueness results use a special structure of the discontinuities as in [14,15,20] or an hyperbolic approach as in [3,13]. We finally remark that problems on network (see [24], [2], [26]) share the same kind of difficulties.…”

Section: Introductionmentioning

confidence: 93%

A Bellman approach for two-domains optimal control problems in ℝ^N

2013

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This article is the starting point of a series of works whose aim is the study of deterministic control problems where the dynamic and the running cost can be completely different in two (or more) complementary domains of the space R N . As a consequence, the dynamic and running cost present discontinuities at the boundary of these domains and this is the main difficulty of this type of problems. We address these questions by using a Bellman approach: our aim is to investigate how to define properly the value function(s), to deduce what is (are) the right Bellman Equation(s) associated to this problem (in particular what are the conditions on the set where the dynamic and running cost are discontinuous) and to study the uniqueness properties for this Bellman equation. In this work, we provide rather complete answers to these questions in the case of a simple geometry, namely when we only consider two different domains which are half spaces: we properly define the control problem, identify the different conditions on the hyperplane where the dynamic and the running cost are discontinuous and discuss the uniqueness properties of the Bellman problem by either providing explicitly the minimal and maximal solution or by giving explicit conditions to have uniqueness.

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Section: Introductionmentioning

confidence: 93%

A Bellman approach for two-domains optimal control problems in ℝ^N

2013

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“…This section is devoted to the proof of a comparison lemma for Hamilton-Jacobi equation for discontinuous super and sub-solutions and for piecewise Lipschitz continuous Hamiltonian. Our proof is inspired by the arguments developed by Ishii [33] and Tourin [51] (see also [2,11,25]). Ishii used a crucial observation of [2] to prove the comparison principle for discontinuous superand sub-soultion of Hamilton-Jacobi equations with nonconvex but continuous Hamiltonian, whereas Tourin gave the uniqueness of continuous solution of Hamilton-Jacobi equations with piecewise Lipschitz continuous Hamiltonian.…”

Section: Appendix a Comparison Principlementioning

confidence: 99%

Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach

Liu¹,

Liu²,

Lam³

2020

Discrete &Amp; Continuous Dynamical Systems - A

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This is part two of our study on the spreading properties of the Lotka-Volterra competition-diffusion systems with a stable coexistence state. We focus on the case when the initial data are exponential decaying. By establishing a comparison principle for Hamilton-Jacobi equations, we are able to apply the Hamilton-Jacobi approach for Fisher-KPP equation due to Freidlin, Evans and Souganidis. As a result, the exact formulas of spreading speeds and their dependence on initial data are derived. Our results indicate that sometimes the spreading speed of the slower species is nonlocally determined. Connections of our results with the traveling profile due to Tang and Fife, as well as the more recent spreading result of Girardin and Lam, will be discussed.

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“…Our results are therefore different from these above. The discontinuity of Hamiltonians we are concerned with is given as a source term instead of the jump of propagating speed, which is also studied recently in [16].…”

Section: H(x P) = −|P| − Ci(x)mentioning

confidence: 99%

Hamilton-Jacobi Equations with Discontinuous Source Terms

Giga

Hamamuki

2012

Communications in Partial Differential Equations

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We study the initial-value problem for a Hamilton-Jacobi equation whose Hamiltonian is discontinuous with respect to state variables. Our motivation comes from a model describing the two dimensional nucleation in crystal growth phenomena. A typical equation has a semicontinuous source term. We introduce a new notion of viscosity solutions and prove among other results that the initial-value problem admits a unique global-in-time uniformly continuous solution for any bounded uniformly continuous initial data. We also give a representation formula of the solution as a value function by the optimal control theory with a semicontinuous running cost function.

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A comparison principle for Hamilton-Jacobi equations with discontinuous Hamiltonians

Abstract: Abstract. We show a comparison principle for viscosity super-and subsolutions to Hamilton-Jacobi equations with discontinuous Hamiltonians. The key point is that the Hamiltonian depends upon u and has a special structure. The supersolution must enjoy some additional regularity.

Cited by 19 publications

References 9 publications

A Bellman approach for two-domains optimal control problems in ℝ^N

A Bellman approach for two-domains optimal control problems in ℝ^N

Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach

Hamilton-Jacobi Equations with Discontinuous Source Terms

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A comparison principle for Hamilton-Jacobi equations with discontinuous Hamiltonians

Abstract: Abstract. We show a comparison principle for viscosity super-and subsolutions to Hamilton-Jacobi equations with discontinuous Hamiltonians. The key point is that the Hamiltonian depends upon u and has a special structure. The supersolution must enjoy some additional regularity.

Cited by 19 publications

References 9 publications

A Bellman approach for two-domains optimal control problems in ℝN

A Bellman approach for two-domains optimal control problems in ℝN

Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach

Hamilton-Jacobi Equations with Discontinuous Source Terms

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A Bellman approach for two-domains optimal control problems in ℝ^N

A Bellman approach for two-domains optimal control problems in ℝ^N