Lipschitz continuity and local semiconcavity for exit time problems with state constraints

Cannarsa, Piermarco; Castelpietra, Marco

doi:10.1016/j.jde.2007.10.020

Cited by 14 publications

(25 citation statements)

References 16 publications

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“…Recall that, if X is compact, then, thanks to Arzelà-Ascoli Theorem [12, Corollary 3, page X. 19], Lip c (X) is compact. For t ∈ R + , we denote by e t : C X → X the evaluation map e t (γ) = γ(t).…”

Section: Notations and Preliminary Definitionsmentioning

confidence: 99%

“…This simplifying assumption is first used when applying Pontryagin Maximum Principle to OCP(X, Γ, k), and, even though it is not strictly needed at this point, as remarked in the beginning of Section 4.3, it is important in Section 4.4 to deduce the semiconcavity of the value function in Proposition 4.18. Indeed, without assumption (e), one must take into account in OCP(X, Γ, k) the state constraint γ(t) ∈ Ω, and value functions of optimal control problems with state constraints may fail to be semiconcave (see, e.g., [19,Example 4.4]).…”

Section: The Mfg Modelmentioning

confidence: 99%

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Minimal-time mean field games

Mazanti

Santambrogio

2019

Math. Models Methods Appl. Sci.

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This paper considers a mean field game model inspired by crowd motion where agents want to leave a given bounded domain through a part of its boundary in minimal time. Each agent is free to move in any direction, but their maximal speed is bounded in terms of the average density of agents around their position in order to take into account congestion phenomena.After a preliminary study of the corresponding minimal-time optimal control problem, we formulate the mean field game in a Lagrangian setting and prove existence of Lagrangian equilibria using a fixed point strategy. We provide a further study of equilibria under the assumption that agents may leave the domain through the whole boundary, in which case equilibria are described through a system of a continuity equation on the distribution of agents coupled with a Hamilton-Jacobi equation on the value function of the optimal control problem solved by each agent. This is possible thanks to the semiconcavity of the value function, which follows from some further regularity properties of optimal trajectories obtained through Pontryagin Maximum Principle. Simulations illustrate the behavior of equilibria in some particular situations.2010 Mathematics Subject Classification. 91A13, 49N70, 49K15, 35Q91.

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Section: Notations and Preliminary Definitionsmentioning

confidence: 99%

Section: The Mfg Modelmentioning

confidence: 99%

Minimal-time mean field games

Mazanti

Santambrogio

2019

Math. Models Methods Appl. Sci.

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“…The asymptotic behavior of solutions to (1) in various settings can be found in [12,13,15,24]. For local semiconcavity of Hamilton-Jacobi equations with state constraints, see [3,20]. There are also various results regarding semiconcavity of different types of equations, for instance, [1,2,6,17,16,20,22].…”

Section: Introductionmentioning

confidence: 99%

Global semiconcavity of solutions to first-order Hamilton-Jacobi equations with state constraints

Han¹

2022

Preprint

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We focus on the global semiconcavity of solutions to first-order Hamilton-Jacobi equations with state constraints, especially for the Hamiltonian H(x, β) := |β| p − f (x) with p ∈ (1, 2]. We first show that the solution is locally semiconcave, and the semiconcavity constant at each point depends on the first time a corresponding minimizing curve emanating from this point hits the boundary. Then, with appropriate conditions on Df , we prove that for any such minimizing curve, the time it takes to hit the boundary of the domain is +∞, and as a consequence, the solution is globally semiconcave. Moreover, the condition on Df is essentially optimal with examples in one-dimensional space. The proofs employ the Euler-Lagrange equations and techniques in weak KAM theory.

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“…The major difficulty in analyzing optimal control problems with state constraints is that their value functions may fail to be semiconcave (see, e.g., [17,Example 4.4]), the latter property being important in the characterization of optimal controls (see, e.g., [18]). In this paper, we rely instead on the techniques introduced in [16] to characterize optimal controls, which do not rely on the semiconcavity of the value function and also allow for weaker regularity assumptions on the dynamics of agents.…”

Section: Introductionmentioning

confidence: 99%

On the characterization of equilibria of nonsmooth minimal-time mean field games with state constraints

Arjmand

Mazanti

2021

2021 60th IEEE Conference on Decision and Control (CDC)

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In this paper, we consider a first-order deterministic mean field game model inspired by crowd motion in which agents moving in a given domain aim to reach a given target set in minimal time. To model interaction between agents, we assume that the maximal speed of an agent is bounded as a function of their position and the distribution of other agents. Moreover, we assume that the state of each agent is subject to the constraint of remaining inside the domain of movement at all times, a natural constraint to model walls, columns, fences, hedges, or other kinds of physical barriers at the boundary of the domain. After recalling results on the existence of Lagrangian equilibria for these mean field games and the main difficulties in their analysis due to the presence of state constraints, we show how recent techniques allow us to characterize optimal controls and deduce that equilibria of the game satisfy a system of partial differential equations, known as the mean field game system.

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Lipschitz continuity and local semiconcavity for exit time problems with state constraints

Cited by 14 publications

References 16 publications

Minimal-time mean field games

Minimal-time mean field games

Global semiconcavity of solutions to first-order Hamilton-Jacobi equations with state constraints

On the characterization of equilibria of nonsmooth minimal-time mean field games with state constraints

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