Full-horizon magnetic control of the Vlasov--Poisson system for plasma confinement
Abstract
We study the full-horizon control of a two-dimensional Vlasov--Poisson system through an externally imposed magnetic field. The control is represented by time-dependent coefficients associated with smooth, spatially localized basis functions, and is designed to steer selected position and velocity moments of the distribution function while limiting particle dispersion and the control effort. Starting from the kinetic constrained optimization problem, we derive the continuous first-order optimality conditions, including a backward Vlasov equation, an elliptic adjoint equation, and the variational inequality associated with bounded controls. We then project this system onto a particle--grid representation and identify the resulting particle adjoints and control gradient. The full-horizon field is then computed on a reduced particle ensemble, initialized by an instantaneous control, and subsequently applied to a larger set of particles. Numerical experiments indicate that the instantaneous control field already provides an effective baseline, while the full-horizon correction can further reduce the relative thermal energy near the boundaries and improve confinement over the prescribed time interval.
Related
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BibTeX
@misc{albi2026fullhorizon,
title = {{Full-horizon magnetic control of the Vlasov--Poisson system for plasma confinement}},
author = {Giacomo Albi and Giacomo Dimarco and Federica Ferrarese and Lorenzo Pareschi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26110},
eprint = {2610.26110},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26110},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Full-horizon magnetic control of the Vlasov--Poisson system for plasma confinement
AU - Giacomo Albi
AU - Giacomo Dimarco
AU - Federica Ferrarese
AU - Lorenzo Pareschi
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26110
UR - https://arxiv.org/abs/2610.26110
AB - We study the full-horizon control of a two-dimensional Vlasov--Poisson system through an externally imposed magnetic field. The control is represented by time-dependent coefficients associated with smooth, spatially localized basis functions, and is designed to steer selected position and velocity moments of the distribution function while limiting particle dispersion and the control effort. Starting from the kinetic constrained optimization problem, we derive the continuous first-order optimality conditions, including a backward Vlasov equation, an elliptic adjoint equation, and the variational inequality associated with bounded controls. We then project this system onto a particle--grid representation and identify the resulting particle adjoints and control gradient. The full-horizon field is then computed on a reduced particle ensemble, initialized by an instantaneous control, and subsequently applied to a larger set of particles. Numerical experiments indicate that the instantaneous control field already provides an effective baseline, while the full-horizon correction can further reduce the relative thermal energy near the boundaries and improve confinement over the prescribed time interval.
ER -