Chapter 22 Differential games

Friedman, Avner

doi:10.1016/s1574-0005(05)80054-2

Cited by 6 publications

(2 citation statements)

References 18 publications

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“…So we require that the parameters α, β, γ satisfy (2.13)-(2. 15), and that f (x, z, p, Γ) satisfy not only the monotonicity condition (1.4) but also the Lipschitz continuity and growth conditions (1.9)-(1.10). To prove that U ε is well defined, we require that the interest rate λ be large enough, condition (1.8), and that h be uniformly bounded.…”

Section: The Elliptic Casementioning

confidence: 99%

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A game interpretation of the Neumann problem for fully nonlinear parabolic and elliptic equations

Daniel

2013

ESAIM: COCV

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Abstract. We provide a deterministic-control-based interpretation for a broad class of fully nonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in a smooth domain. We construct families of two-person games depending on a small parameter ε which extend those proposed by Kohn and Serfaty [21]. These new games treat a Neumann boundary condition by introducing some specific rules near the boundary. We show that the value function converges, in the viscosity sense, to the solution of the PDE as ε tends to zero. Moreover, our construction allows us to treat both the oblique and the mixed type Dirichlet-Neumann boundary conditions.

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Section: The Elliptic Casementioning

confidence: 99%

“…In calling this Helen's value function, we are using an established convention from the theory of discrete-time, two person games (see e.g. [15]).…”

Section: 1mentioning

confidence: 99%

A game interpretation of the Neumann problem for fully nonlinear parabolic and elliptic equations

Daniel

2013

ESAIM: COCV

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A deterministic‐control‐based approach to fully nonlinear parabolic and elliptic equations

Kohn

Serfaty

2010

Comm Pure Appl Math

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We show that a broad class of fully nonlinear, second-order parabolic or elliptic PDEs can be realized as the Hamilton-Jacobi-Bellman equations of deterministic two-person games. More precisely: given the PDE, we identify a deterministic, discrete-time, two-person game whose value function converges in the continuous-time limit to the viscosity solution of the desired equation. Our game is, roughly speaking, a deterministic analogue of the stochastic representation recently introduced by Cheridito, Soner, Touzi, and Victoir. In the parabolic setting with no u-dependence, it amounts to a semidiscrete numerical scheme whose timestep is a min-max. Our result is interesting, because the usual control-based interpretations of second-order PDEs involve stochastic rather than deterministic control.

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Review of Maneuvering Strategies in the Level Flight Section of UAV

Sun,

Hao

2024

Lecture Notes in Electrical Engineering

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Chapter 22 Differential games

Cited by 6 publications

References 18 publications

A game interpretation of the Neumann problem for fully nonlinear parabolic and elliptic equations

A game interpretation of the Neumann problem for fully nonlinear parabolic and elliptic equations

A deterministic‐control‐based approach to fully nonlinear parabolic and elliptic equations

Review of Maneuvering Strategies in the Level Flight Section of UAV

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