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Abstract We consider an optimal control problem where the state equations are a coupled hyperbolic–elliptic system. This system arises in elastodynamics with piezoelectric effects—the elastic stress tensor is a function of elastic displacement and electric potential. The electric flux acts as the control variable and bound constraints on the control are considered. We develop a complete analysis for the state equations and the control problem. The requisite regularity on the control, to show the well-posedness of the state equations, is enforced using the cost functional. We rigorously derive the first-order necessary and sufficient conditions using adjoint equations and further study their well-posedness. For spatially discrete (time-continuous) problems, we show the convergence of our numerical scheme. Three-dimensional numerical experiments are provided showing convergence properties of a fully discrete method and the practical applicability of our approach.
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This content will become publicly available on June 13, 2026
Kinetic‐type mean field games with non‐separable local Hamiltonians
Abstract We prove well‐posedness of a class ofkinetic‐typemean field games (MFGs), which typically arise when agents control their acceleration. Such systems include independent variables representing the spatial position as well as velocity. We consider non‐separable Hamiltonians without any structural conditions, which depend locally on the density variable. Our analysis is based on two main ingredients: an energy method for the forward–backward system in Sobolev spaces, on the one hand, and on a suitablevector field methodto control derivatives with respect to the velocity variable, on the other hand. The careful combination of these two techniques reveals interesting phenomena applicable for MFGs involving general classes of drift‐diffusion operators and non‐linearities. While many prior existence theories for general MFGs systems take the final datum function to be smoothing, we can allow this function to be non‐smoothing, that is, also depending locally on the final measure. Our well‐posedness results hold under an appropriate smallness condition, assumed jointly on the data.
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- Award ID(s):
- 2307638
- PAR ID:
- 10608271
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Journal of the London Mathematical Society
- Volume:
- 111
- Issue:
- 6
- ISSN:
- 0024-6107
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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