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A finite horizon nonlinear optimal control problem is considered for which the associated Hamiltonian satisfies a uniform semiconcavity property with respect to its state and costate variables. It is shown that the value function for this optimal control problem is equivalent to the value of a min-max game, provided the time horizon considered is sufficiently short. This further reduces to maximization of a linear functional over a convex set. It is further proposed that the min-max game can be relaxed to a type of stat (stationary) game, in which no time horizon constraint is involved.
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This content will become publicly available on July 1, 2025
Lagrangian, game theoretic, and PDE methods for averaging G-equations in turbulent combustion: existence and beyond
G-equations are popular level set Hamilton–Jacobi nonlinear partial differential equations (PDEs) of first or second order arising in turbulent combustion. Characterizing the effective burning velocity (also known as the turbulent burning velocity) is a fundamental problem there. We review relevant studies of the G-equation models with a focus on both the existence of effective burning velocity (homogenization), and its dependence on physical and geometric parameters (flow intensity and curvature effect) through representative examples. The corresponding physical background is also presented to provide motivations for mathematical problems of interest. Thelack of coercivityof Hamiltonian is a hallmark of G-equations. When either the curvature of the level set or the strain effect of fluid flows is accounted for, the Hamiltonian becomeshighly nonconvex and nonlinear. In the absence of coercivity and convexity, the PDE (Eulerian) approach suffers from insufficient compactness to establish averaging (homogenization). We review and illustrate a suite of Lagrangian tools, most notably min-max (max-min) game representations of curvature and strain G-equations, working in tandem with analysis of streamline structures of fluid flows and PDEs. We discuss open problems for future development in this emerging area of dynamic game analysis for averaging noncoercive, nonconvex, and nonlinear PDEs such as geometric (curvature-dependent) PDEs with advection.
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- PAR ID:
- 10516977
- Publisher / Repository:
- American Mathematical Society
- Date Published:
- Journal Name:
- Bulletin of the American Mathematical Society
- Volume:
- 61
- Issue:
- 3
- ISSN:
- 0273-0979
- Page Range / eLocation ID:
- 470 to 514
- Subject(s) / Keyword(s):
- Hamilton–Jacobi PDEs noncoercivity nonconvexity Lagrangian and game methods averaging turbulent combustion.
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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