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Optimal Control of H-Mode Tokamak Plasma Temperature based on Pontryagin's Principle

Slim Jmal, Matteo Tacchi-Bénard, Emmanuel Witrant

TL;DR

This work develops a forward-looking, adjoint-based optimal control framework for Tokamak plasma temperature, casting the problem as a PDE-constrained optimization with a continuum of receding horizons. By coupling a nonlinear diffusion model for $T_e(x,t)$ with an extended Bohm/gyro-Bohm diffusivity and a receding-horizon Pontryagin principle, the authors derive a forward-in-time feedback controller via a state–costate system and an adaptive regularization parameter $α(t)$. They establish regularity and convergence properties, and validate the approach on Tore Supra data, showing exponential error decay and bounded control energy while achieving near-perfect tracking of the target profile. The methodology bridges PMP theory and model-predictive control in infinite-dimensional settings, offering a scalable, real-time capable tool for achieving robust, energy-efficient high-confinement operation in current and future fusion devices.

Abstract

This paper studies the decay of an objective functional using a new control technique within Pontryagin's framework. Convergence analysis is carried out on the infinite-dimensional space of Tokamak plasma dynamical state as described by weakly decoupled nonlinear partial differential equations. An adjoint-based optimal control is derived to minimize the deviation from a predefined dynamical trajectory leading to the desired target state at stationary regime, by turning Pontryagin's transversality conditions into a continuum of horizons. A feedback controller is proposed to steer the system efficiently in real time, as opposed to an open-loop controller resulting from the classical Pontryagin's setting. An algorithm synthesizing the constraint-free optimal controller is used for profile tracking based on experimental data.

Optimal Control of H-Mode Tokamak Plasma Temperature based on Pontryagin's Principle

TL;DR

This work develops a forward-looking, adjoint-based optimal control framework for Tokamak plasma temperature, casting the problem as a PDE-constrained optimization with a continuum of receding horizons. By coupling a nonlinear diffusion model for with an extended Bohm/gyro-Bohm diffusivity and a receding-horizon Pontryagin principle, the authors derive a forward-in-time feedback controller via a state–costate system and an adaptive regularization parameter . They establish regularity and convergence properties, and validate the approach on Tore Supra data, showing exponential error decay and bounded control energy while achieving near-perfect tracking of the target profile. The methodology bridges PMP theory and model-predictive control in infinite-dimensional settings, offering a scalable, real-time capable tool for achieving robust, energy-efficient high-confinement operation in current and future fusion devices.

Abstract

This paper studies the decay of an objective functional using a new control technique within Pontryagin's framework. Convergence analysis is carried out on the infinite-dimensional space of Tokamak plasma dynamical state as described by weakly decoupled nonlinear partial differential equations. An adjoint-based optimal control is derived to minimize the deviation from a predefined dynamical trajectory leading to the desired target state at stationary regime, by turning Pontryagin's transversality conditions into a continuum of horizons. A feedback controller is proposed to steer the system efficiently in real time, as opposed to an open-loop controller resulting from the classical Pontryagin's setting. An algorithm synthesizing the constraint-free optimal controller is used for profile tracking based on experimental data.

Paper Structure

This paper contains 14 sections, 2 theorems, 47 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Description and Control Problem
  3. Electron Temperature Dynamics
  4. Electron Heat Diffusivity Model
  5. Optimal Control Problem
  6. A Pontryagin-Based Approach
  7. Receding Horizon Pontryagin's Principle
  8. Regularity Analysis
  9. Convergence Analysis
  10. Control Algorithm Implementation and Simulation Results
  11. Control Algorithm
  12. Simulation Results
  13. Code Script
  14. Conclusion

Key Result

Theorem 1

Let $w \in L^\infty(\Omega)$ be a nonnegative function such that $w(x) > 0$ a.e.. Then there exists a constant $C > 0$ such that for all $v \in H^1_w(\Omega)$ The optimal Poincaré constant is $C = \lambda_{1,w}(\Omega)^{-1}$, where $\lambda_{1,w}$ is the first Dirichlet-Laplacian eigenvalue of $\Omega$ as the infimum of the weighted Rayleigh quotient $\mathcal{R}_w$ over $H^1_w(\Omega)$ and $\| \c

Figures (6)

  • Figure 1: Boundedness of the Optimal Power Input
  • Figure 2: Controlled Plasma Temperature Evolution
  • Figure 3: Controlled Plasma Temperature at Final Time
  • Figure 4: Optimal Power Input Control Law
  • Figure 5: Convergence of the Objective Functional
  • ...and 1 more figures

Theorems & Definitions (5)

  • Remark 1
  • Theorem 1: Weighted Poincaré Inequality
  • Remark 2
  • Lemma 1.1: Gradient Flow
  • Remark 3