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On the stabilization of a kinetic model by feedback-like control fields in a Monte Carlo framework

Jan Bartsch, Alfio Borzi

Abstract

The construction of feedback-like control fields for a kinetic model in phase space is investigated. The purpose of these controls is to drive an initial density of particles in the phase space to reach a desired cyclic trajectory and follow it in a stable way. For this purpose, an ensemble optimal control problem governed by the kinetic model is formulated in a way that is amenable to a Monte Carlo approach. The proposed formulation allows to define a one-shot solution procedure consisting in a backward solve of an augmented adjoint kinetic model. Results of numerical experiments demonstrate the effectiveness of the proposed control strategy.

On the stabilization of a kinetic model by feedback-like control fields in a Monte Carlo framework

Abstract

The construction of feedback-like control fields for a kinetic model in phase space is investigated. The purpose of these controls is to drive an initial density of particles in the phase space to reach a desired cyclic trajectory and follow it in a stable way. For this purpose, an ensemble optimal control problem governed by the kinetic model is formulated in a way that is amenable to a Monte Carlo approach. The proposed formulation allows to define a one-shot solution procedure consisting in a backward solve of an augmented adjoint kinetic model. Results of numerical experiments demonstrate the effectiveness of the proposed control strategy.
Paper Structure (8 sections, 2 theorems, 45 equations, 6 figures, 1 table, 6 algorithms)

This paper contains 8 sections, 2 theorems, 45 equations, 6 figures, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. A kinetic optimal control problem
  3. Optimality system
  4. A Monte Carlo approach
  5. Monte Carlo simulation of the kinetic model
  6. MC simulation of the adjoint kinetic model
  7. Numerical experiments
  8. Conclusion

Key Result

Theorem 2.1

Assume that asmpt:Existence_Assumption_VanDerMee holds and suppose $1\leq p < \infty$ and $0 \leq \alpha < 1$. Furthermore, assume that and $f_0 \in L^p(\Omega \times \mathbb{R}^d)$. Then there exists a unique $f \in L^p({\mathscr{Q}})$ that solves eq:controlled_Init_bdry_value_problem. Furthermore,

Figures (6)

  • Figure 1: Quiver plot of the calculated control. The solid ellipse is the curve $z_D(t)$, $t\in[0,T]$. The arrows are given by the scaled vector $(v,u(x,v,t))^T$.
  • Figure 2: Evolution of $f$ starting from a uniform initial distribution and subject to the control field $u$.
  • Figure 3: Control field and corresponding distribution of particles at final time using different values of the denoising parameter $c_s$.
  • Figure 4: Control field at final time for different values of the control weight $\nu$.
  • Figure 5: Averaged control $\bar{u}$ defined in \ref{['eControlStat']}; quiver and 3D plot.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Remark 4.1