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Instantaneous control strategies for magnetically confined fusion plasma

Giacomo Albi, Giacomo Dimarco, Federica Ferrarese, Lorenzo Pareschi

TL;DR

The paper develops an instantaneous feedback control framework for magnetically confined plasma described by the kinetic Vlasov-Poisson system with an external magnetic field. It formulates and implements both discretize-then-optimize and optimize-then-discretize strategies, integrated with particle-in-cell numerics to produce per-cell magnetic-field controls that steer the plasma away from walls. Through exact cellwise feedback laws and adjoint-based variants, the work demonstrates consistency between the two approaches in the continuous limit and validates the method with two-stream and Kelvin-Helmholtz instability tests, showing reduced wall interaction and lower boundary energy. The results highlight the potential of fast, localized magnetic controls for confinement in fusion-relevant kinetic models and point to future extensions to full 3D Maxwell coupling, collisions, and uncertainty handling.

Abstract

The principle behind magnetic fusion is to confine high temperature plasma inside a device in such a way that the nuclei of deuterium and tritium joining together can release energy. The high temperatures generated needs the plasma to be isolated from the wall of the device to avoid damages and the scope of external magnetic fields is to achieve this goal. In this paper, to face this challenge from a numerical perspective, we propose an instantaneous control mathematical approach to steer a plasma into a given spatial region. From the modeling point of view, we focus on the Vlasov equation in a bounded domain with self induced electric field and an external strong magnetic field. The main feature of the control strategy employed is that it provides a feedback on the equation of motion based on an instantaneous prediction of the discretized system. This permits to directly embed the minimization of a given cost functional into the particle interactions of the corresponding Vlasov model. The numerical results demonstrate the validity of our control approach and the capability of an external magnetic field, even if in a simplified setting, to lead the plasma far from the boundaries.

Instantaneous control strategies for magnetically confined fusion plasma

TL;DR

The paper develops an instantaneous feedback control framework for magnetically confined plasma described by the kinetic Vlasov-Poisson system with an external magnetic field. It formulates and implements both discretize-then-optimize and optimize-then-discretize strategies, integrated with particle-in-cell numerics to produce per-cell magnetic-field controls that steer the plasma away from walls. Through exact cellwise feedback laws and adjoint-based variants, the work demonstrates consistency between the two approaches in the continuous limit and validates the method with two-stream and Kelvin-Helmholtz instability tests, showing reduced wall interaction and lower boundary energy. The results highlight the potential of fast, localized magnetic controls for confinement in fusion-relevant kinetic models and point to future extensions to full 3D Maxwell coupling, collisions, and uncertainty handling.

Abstract

The principle behind magnetic fusion is to confine high temperature plasma inside a device in such a way that the nuclei of deuterium and tritium joining together can release energy. The high temperatures generated needs the plasma to be isolated from the wall of the device to avoid damages and the scope of external magnetic fields is to achieve this goal. In this paper, to face this challenge from a numerical perspective, we propose an instantaneous control mathematical approach to steer a plasma into a given spatial region. From the modeling point of view, we focus on the Vlasov equation in a bounded domain with self induced electric field and an external strong magnetic field. The main feature of the control strategy employed is that it provides a feedback on the equation of motion based on an instantaneous prediction of the discretized system. This permits to directly embed the minimization of a given cost functional into the particle interactions of the corresponding Vlasov model. The numerical results demonstrate the validity of our control approach and the capability of an external magnetic field, even if in a simplified setting, to lead the plasma far from the boundaries.
Paper Structure (13 sections, 1 theorem, 79 equations, 13 figures)

This paper contains 13 sections, 1 theorem, 79 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Particle-in-cell (PIC) Methods
  4. Derivation of the instantaneous control strategy
  5. Discretize then optimize (DtO)
  6. Optimize then discretize (OtD)
  7. Numerical experiments
  8. Two stream plasma test case
  9. Controlling the velocity and the position field.
  10. Kelvin-Helmholtz instability.
  11. Conclusions
  12. Diocotron instability: dynamics.
  13. Diocotron instability: numerical errors.

Key Result

Proposition 1

Assume the parameters to scale as then the feedback control at cell $C_k$ associated to eq:discr_J reads as follows where $\gamma>0$, and with $\mathbb{P}_{[-M,M]}(\cdot)$ denoting the projection over the interval $[-M,M]$. In the limit $h\to 0$ the control at the continuous level reads, with

Figures (13)

  • Figure 1: On the left the three dimensional torus, on the right the simplified geometry considered.
  • Figure 2: Two stream plasma test. Initial density top left. Initial thermal energy (top right). Initial electric field (bottom left) and velocity (bottom right) in the $y$-direction.
  • Figure 3: Two stream plasma test with constant magnetic field $B=1$. First row: mass density taken at time $t=5$, $t =50$ and $t=100$. Second row: mean velocity in the $y$-direction (on the left), electric field in the $y$-direction (on the right).
  • Figure 4: Two stream plasma test with constant magnetic field $B=1$. Density percentages defined as in equation \ref{['eq:mass_boundary']} (on the left). Thermal energy at the boundaries defined as in \ref{['eq:energy']} (on the right).
  • Figure 5: Two stream plasma test with control. We set $\alpha_\emph{x} = \alpha_\emph{v} = 1.5$, $\beta_\emph{x} =\beta_\emph{v}= 0.1$, and $\gamma = 10^{-3}$. First row: mass density taken at time $t=5$, $t =50$ and $t=100$. Second row: mean velocity in the $y$-direction (on the left), electric field in the $y$-direction (in the middle), magnetic field (on the right).
  • ...and 8 more figures

Theorems & Definitions (5)

  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Remark 3