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Optimal control of a kinetic equation

Aaron Pim, Tristan Pryer, Alex Trenam

TL;DR

This work addresses PDE-constrained optimal control with a degenerate kinetic Kolmogorov constraint $u_t + v u_x - \epsilon u_{vv} = f$. It develops a hypocoercivity framework to obtain well-posedness and decay estimates for both primal and adjoint problems and introduces a hypocoercivity-preserving finite element discretisation. The methodology yields discrete coercivity and best-approximation properties, validated by numerical experiments across stationary and time-dependent, space-time, and constrained settings. The results provide robust tools for long-time simulations in kinetic PDE-constrained optimization and point to future extensions to Boltzmann–Fokker–Planck models and related kinetic systems.

Abstract

This work addresses an optimal control problem constrained by a degenerate kinetic equation of parabolic-hyperbolic type. Using a hypocoercivity framework we establish the well-posedness of the problem and demonstrate that the optimal solutions exhibit a hypocoercive decay property, ensuring stability and robustness. Building on this framework, we develop a finite element discretisation that preserves the stability properties of the continuous system. The effectiveness and accuracy of the proposed method are validated through a series of numerical experiments, showcasing its ability to handle challenging PDE-constrained optimal control problems.

Optimal control of a kinetic equation

TL;DR

This work addresses PDE-constrained optimal control with a degenerate kinetic Kolmogorov constraint . It develops a hypocoercivity framework to obtain well-posedness and decay estimates for both primal and adjoint problems and introduces a hypocoercivity-preserving finite element discretisation. The methodology yields discrete coercivity and best-approximation properties, validated by numerical experiments across stationary and time-dependent, space-time, and constrained settings. The results provide robust tools for long-time simulations in kinetic PDE-constrained optimization and point to future extensions to Boltzmann–Fokker–Planck models and related kinetic systems.

Abstract

This work addresses an optimal control problem constrained by a degenerate kinetic equation of parabolic-hyperbolic type. Using a hypocoercivity framework we establish the well-posedness of the problem and demonstrate that the optimal solutions exhibit a hypocoercive decay property, ensuring stability and robustness. Building on this framework, we develop a finite element discretisation that preserves the stability properties of the continuous system. The effectiveness and accuracy of the proposed method are validated through a series of numerical experiments, showcasing its ability to handle challenging PDE-constrained optimal control problems.

Paper Structure

This paper contains 15 sections, 24 theorems, 159 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Notation and problem setup
  3. Hypocoercivity for the Kolmogorov equation
  4. The stationary Kolmogorov operator
  5. The time dependent Kolmogorov equation
  6. Optimal control
  7. Stationary control
  8. Space-time control
  9. Hypocoercivity-preserving finite element methods
  10. Stationary Kolmogorov
  11. Spatial semi-discretisation of the time-dependent Kolmogorov
  12. Stationary Kolmogorov constrained optimal control
  13. Spatial semi-discretisation of time-dependent Kolmogorov constrained optimal control
  14. Numerical results
  15. Concluding remarks

Key Result

Lemma 3.1

For $w\in\operatorname H\xspace^{1}(\Omega)$, we have where $C^-_{\rm{eq}} := \min\!\left({1, \lambda_-\!\left({\boldsymbol{M}}\right)}\right)$ and $C^+_{\rm{eq}} := \max\!\left({1, \lambda_+\!\left({\boldsymbol{M}}\right)}\right)$. Thus if $\boldsymbol{M}$ is strictly positive definite, then the $\|\cdot\|_{\mathcal{H}}$ and $\|\cdot\|_{\operatorname

Figures (11)

  • Figure 1: An illustration of the domain $\Omega$ and the relevant boundary regions.
  • Figure 2: Plots of the primal problem solution errors for the benchmark tests in Example \ref{['ex:forward-problem-benchmark']}, using polynomial degrees $r = 2, 3, 4$. Straight dotted lines are plotted, along with their gradients, to aid in demonstrating the order of convergence.
  • Figure 3: Plots of the primal and dual problem solution errors, measured in the $\mathcal{H}$ norm, for the benchmark tests in Example \ref{['ex:optimal-control-benchmark']}, using polynomial degrees $r = 2, 3, 4$. Straight dotted lines are plotted, along with their gradients, to aid in demonstrating the order of convergence.
  • Figure 4: Target function profiles for the physically-motivated experiments in Example \ref{['ex:alpha-dependence']}.
  • Figure 5: Profiles of the primal (top) and control (bottom) variables for various values of $\alpha$ in the large diffusion regime ($\epsilon = 10^{-1}$), using the first target function $\mathcal{D}_1$ in Example \ref{['ex:alpha-dependence']}.
  • ...and 6 more figures

Theorems & Definitions (61)

  • Remark 2.1: Connection to Boltzmann transport
  • Lemma 3.1: Norm equivalence
  • proof
  • Remark 3.2: Alternative interpretation
  • Remark 3.3: Boundary conditions
  • Lemma 3.4: Coercivity of the bilinear form
  • proof
  • Lemma 3.5: Continuous dependence on problem data
  • proof
  • Remark 3.6: A specific regularisation
  • ...and 51 more