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

*(English)*Zbl 1204.35070The paper shows that a broad class of fully nonlinear, second-order parabolic or elliptic distributed parameter systems (PDEs) can be realized as the Hamilton-Jacobi-Bellman equations of deterministic two-person games. Namely, given a PDE, one can 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. This game is a deterministic analogue of the corresponding stochastic game. In the parabolic setting with no \(u\)-dependence, it amounts to a semidiscrete numerical scheme whose time step is a min-max. The results of the work are quite interesting, because the usual control-based interpretations of second-order PDEs involve stochastic rather than deterministic control.

Reviewer: Vyacheslav I. Maksimov (Ekaterinburg)

##### MSC:

35F21 | Hamilton-Jacobi equations |

35Q93 | PDEs in connection with control and optimization |

91A05 | 2-person games |

35K55 | Nonlinear parabolic equations |

35J60 | Nonlinear elliptic equations |

91A50 | Discrete-time games |

##### Keywords:

Hamilton-Jacobi-Bellman equation; deterministic game; viscosity solution; semidiscrete numerical scheme
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\textit{R. V. Kohn} and \textit{S. Serfaty}, Commun. Pure Appl. Math. 63, No. 10, 1298--1350 (2010; Zbl 1204.35070)

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