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

  • *Corresponding author: Jan Bartsch

    *Corresponding author: Jan Bartsch 

The first author is partially supported by Deutsche Forschungsgemeinschaft (DFG) within SFB 1432, Project-ID 425217212

Abstract Full Text(HTML) Figure(6) / Table(1) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 49M05, 49M41, 65C05, 65K10.


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  • 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 (17); quiver and 3D plot

    Figure 6.  Evolution of $ f $, starting with an initial Gaussian distribution and subject to the averaged control $ \bar{u} $. Time ordering as in Figure 1

    Table 1.  Numerical and physical parameters

    Symbol Value Symbol Value
    $ N_t $ 100 $ \Delta t $ $ 0.025 $
    $ N_x \times N_v $ [-] $ 50 \times 50 $ $ v_{\max} $ $ 5 $
    $ p_{\max} $ $ 10.0 $ $ \Delta v $ $ 0.2 $
    $ \Delta p $ $ 0.2 $ $ N_f $ $ 10^4 $
    $ \gamma $ $ 0.9999 $ $ \alpha $ $ 0.5 $
    $ \nu $ $ 1 $ $ C_\theta $, $ C_\varphi $ $ 10^3 $
    $ c_s $ $ 0.5 $ $ N_q^{N_t} $ $ 6 \cdot 10^2 $
     | Show Table
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