Viscosity solutions for junctions: well posedness and stability

Lions, Pierre-Louis; Souganidis, Panagiotis E.

doi:10.4171/rlm/747

Cited by 36 publications

(61 citation statements)

References 7 publications

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“…Preliminaries. We begin by recalling the definition of viscosity solutions of (1) from [LS1]. An upper (resp.…”

Section: Time-independent Problem: Vanishing Viscosity Limitmentioning

confidence: 99%

“…Previous Work. The well-posedness of (1) and (2) was recently established by Lions and Souganidis in [LS1,LS2]. In addition to establishing comparison for these equations for general (non-convex) Hamiltonians, they showed that HJ equations with Kirchoff junction conditions include as a special case the so-called flux-limited Hamilton-Jacobi equations introduced by Imbert and Monneau [IM] in the setting of convex and quasi-convex Hamiltonians.…”

Section: Introductionmentioning

confidence: 96%

“…The error analysis of approximations of (1) relies on an insight from [LS1]. Specifically, if u is a function on [−1, 0], u ′ (0) exists, θ, δ > 0 are parameters, and y ∈ [−1, 0], then the function…”

Section: Introductionmentioning

confidence: 99%

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Convergence & rates for Hamilton–Jacobi equations with Kirchoff junction conditions

Morfe

2020

Nonlinear Differ. Equ. Appl.

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We obtain rates of convergence for two approximation schemes of timeindependent and time-dependent Hamilton-Jacobi equations on networks with Kirchoff junction conditions. We analyze the vanishing viscosity limit and monotone finite-difference schemes. Following recent work of Lions and Souganidis, we impose no convexity assumptions on the Hamiltonians. For stationary Hamilton-Jacobi equations, we obtain the optimal ǫ 1 2 rate, while we obtain an ǫ 1 6 rate for approximations of the Cauchy problem. In addition, we present a number of new techniques of independent interest, including a simplified comparison proof for the Cauchy problem and an equivalent definition of the Kirchoff junction condition.

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“…Preliminaries. We begin by recalling the definition of viscosity solutions of (1) from [LS1]. An upper (resp.…”

Section: Time-independent Problem: Vanishing Viscosity Limitmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

See 1 more Smart Citation

Convergence & rates for Hamilton–Jacobi equations with Kirchoff junction conditions

Morfe

2020

Nonlinear Differ. Equ. Appl.

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“…We suggest two different proofs of the comparison principle. The first one is inspired from the work by Lions & Souganidis [21] and uses arguments from the theory of PDEs, and the second one uses a blend of arguments from optimal control theory and PDE techniques suggested in [7,8,3] and [25]. Finally, in Section 4, the same program is carried out when the strong controllability is replaced by the weaker one that we coin 'moderate controllability near the interface'.…”

Section: Introductionmentioning

confidence: 99%

Hamilton–Jacobi equations for optimal control on multidimensional junctions with entry costs

Dao

Djehiche

2020

Nonlinear Differ. Equ. Appl.

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We consider an infinite horizon control problem for dynamics constrained to remain on a multidimensional junction with entry costs. We derive the associated system of Hamilton-Jacobi equations (HJ), prove the comparison principle and that the value function of the optimal control problem is the unique viscosity solution of the HJ system. This is done under the usual strong controllability assumption and also under a weaker condition, coined 'moderate controllability assumption'.

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“…This will show that the functions (v i ) 1≤i≤N are (suitably defined) viscosity solutions of the following system • The term H T O accounts for situations in which the trajectory stays at O. The most important part of the paper will be devoted to two different proofs of a comparison principle leading to the well-poseness of (1.1): the first one uses arguments from optimal control theory coming from Barles et al [6,7] and Achdou et al [3]; the second one is inspired by Lions and Souganidis [19] and uses arguments from the theory of PDEs.…”

Section: Introductionmentioning

confidence: 99%

Hamilton-Jacobi equations for optimal control on networks with entry or exit costs

Dao

2019

ESAIM: COCV

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We consider an optimal control on networks in the spirit of the works of Achdou et al. (2013) and Imbert et al. (2013). The main new feature is that there are entry (or exit) costs at the edges of the network leading to a possible discontinuous value function. We characterize the value function as the unique viscosity solution of a new Hamilton-Jacobi system. The uniqueness is a consequence of a comparison principle for which we give two different proofs, one with arguments from the theory of optimal control inspired by Achdou et al. (2014) and one based on partial differential equations techniques inspired by a recent work of Lions and Souganidis (2016).The first articles about optimal control problems in which the set of admissible states is a network (therefore the state variable is a continuous one) appeared in 2012: in [2], Achdou et al. derived the HJB equation associated to an infinite horizon optimal control on a network and proposed a suitable notion of viscosity solution. Obviously, the main difficulties arise at the vertices where the network does not have a regular differential structure. As a result, the new admissible test-functions whose restriction to each edge is C 1 are applied. Independently and at the same time, Imbert et al. [17] proposed an equivalent notion of viscosity solution for studying a Hamilton-Jacobi approach to junction problems and traffic flows. Both [2] and [17] contain first results on comparison principles which were improved later. It is also worth mentioning the work by Schieborn and Camilli [22], in which the authors focus on eikonal equations on networks and on a less general notion of viscosity solution. In the particular case of eikonal equations, Camilli and Marchi established in [10] the equivalence between the definitions given in [2,17,22].Since 2012, several proofs of comparison principles for HJB equations on networks, giving uniqueness of the solution, have been proposed. In [3], Achdou et al. give a proof of a comparison principle for a stationary HJB equationarising from an optimal control with infinite horizon, (therefore the Hamiltonian is convex) by mixing arguments from the theory of optimal control and PDE techniques. Such a proof was much inspired by works of Barles et al. [7,6], on regional optimal control problems in R d , (with discontinuous dynamics and costs). 2.A different and more general proof, using only arguments from the theory of PDEs was obtained by Imbert and Monneau in [16]. The proof works for quasi-convex Hamiltonians, and for stationary and time-dependent HJB equations. It relies on the construction of suitable vertex test functions. 3.A very simple and elegant proof, working for non convex Hamiltonians, has been very recently given by Lions and Souganidis [19,20].

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Viscosity solutions for junctions: well posedness and stability

Cited by 36 publications

References 7 publications

Convergence & rates for Hamilton–Jacobi equations with Kirchoff junction conditions

Convergence & rates for Hamilton–Jacobi equations with Kirchoff junction conditions

Hamilton–Jacobi equations for optimal control on multidimensional junctions with entry costs

Hamilton-Jacobi equations for optimal control on networks with entry or exit costs

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