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

Jan Bartsch; Alfio Borzì

doi:10.3934/krm.2024005

Article Contents

2024, Volume 17, Issue 6: 892-913. Doi: 10.3934/krm.2024005

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

Jan Bartsch^1, , and
Alfio Borzì^2,

1.
Fachbereich für Mathematik und Statistik, Universität Konstanz, Universitätsstr. 10, 78464 Konstanz, Germany
2.
Institut für Mathematik, Universität Würzburg, Emil-Fischer-Str. 30, 97074 Würzburg, Germany

^*Corresponding author: Jan Bartsch
^*Corresponding author: Jan Bartsch

Received: May 2023

Revised: December 2023

Early access: February 2024

Published: December 2024

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

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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.

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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 $

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Figure 2. Evolution of $ f $ starting from a uniform initial distribution and subject to the control field $ u $

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Figure 3. Control field and corresponding distribution of particles at final time using different values of the denoising parameter $ c_s $

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Figure 4. Control field at final time for different values of the control weight $ \nu $

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Figure 5. Averaged control $ \bar{u} $ defined in (17); quiver and 3D plot

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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

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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 $

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References

[1]	G. Albi, Y.-P. Choi, M. Fornasier and D. Kalise, Mean field control hierarchy, Appl. Math. Optim., 76 (2017), 93-135. doi: 10.1007/s00245-017-9429-x.
[2]	J. Bartsch and A. Borzì, MOCOKI: A Monte Carlo approach for optimal control in the force of a linear kinetic model, Comput. Phys. Commun., 266 (2021), 108030, 13 pp. doi: 10.1016/j.cpc.2021.108030.
[3]	J. Bartsch, A. Borzì, F. Fanelli and S. Roy, A theoretical investigation of Brockett's ensemble optimal control problems, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 162, 34 pp. doi: 10.1007/s00526-019-1604-2.
[4]	J. Bartsch, G. Nastasi and A. Borzì, Optimal control of the Keilson-Storer master equation in a Monte Carlo framework, J. Comput. Theor. Transp., 50 (2021), 454-482. doi: 10.1080/23324309.2021.1896552.
[5]	R. Beals and V. Protopopescu, Abstract time-dependent transport equations, J. Math. Anal. Appl., 121 (1987), 370-405. doi: 10.1016/0022-247X(87)90252-6.
[6]	R. Bellman, Dynamic Programming, Princeton University Press, 1957.
[7]	P. R. Berman, J. E. M. Haverkort and J. P. Woerdman, Collision kernels and transport coefficients, Phys. Rev. A, 34 (1986), 4647-4656.
[8]	P. Berman and V. Malinovsky, Principles of Laser Spectroscopy and Quantum Optics, Princeton University Press, 2010.
[9]	G. E. P. Box and M. E. Muller, A note on the generation of random normal deviates, Ann. Math. Statist., 29 (1958), 610-611.
[10]	R. W. Brockett, Minimum attention control, Proceedings of the 36th IEEE Conference on Decision and Control, 3 (1997), 2628-2632.
[11]	R. W. Brockett, Optimal Control of the Liouville Equation, In AMS/IP Stud. Adv. Math., 39 American Mathematical Society, Providence, RI, 2007. doi: 10.1090/amsip/039/02.
[12]	R. W. Brockett, Notes on the control of the Liouville equation, Control of Partial Differential Equations, 2048 (2012), 101-129. doi: 10.1007/978-3-642-27893-8_2.
[13]	R. Caflisch, D. Silantyev and Y. Yang, Adjoint DSMC for nonlinear Boltzmann equation constrained optimization, J. Comput. Phys., 439 (2021), Paper No. 110404, 29 pp. doi: 10.1016/j.jcp.2021.110404.
[14]	J. Chen and M.-Z. Yang, Linear transport equation with specular reflection boundary condition, Transport Theory Statist. Phys., 20 (1991), 281-306. doi: 10.1080/00411459108203907.
[15]	G. Fabbri, F. Gozzi and A. Świech, Stochastic Optimal Control in Infinite Dimension: Dynamic Programming and HJB Equations, Probability Theory and Stochastic Modelling. Springer International Publishing, 2017. doi: 10.1007/978-3-319-53067-3.
[16]	M. F. Gelin, A. P. Blokhin, V. A. Tolkachev and W. Domcke, Microscopic derivation of the Keilson - Storer master equation, J. Chem. Phys., 462 (2015), 35-40.
[17]	M. F. Gelin and D. S. Kosov, Molecular reorientation in hydrogen-bonding liquids: Through algebraic t - 3/2 relaxation toward exponential decay, J. Chem. Phys., 124 (2006), 144514.
[18]	E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration illustrated by the Störmer-Verlet method, Acta Numer., 12 (2003), 399-450. doi: 10.1017/S0962492902000144.
[19]	M. B. Horowitz, A. Damle and J. W. Burdick, Linear hamilton jacobi bellman equations in high dimensions, In 53rd IEEE Conference on Decision and Control, IEEE, (2014), 5880-5887.
[20]	C. Jacoboni and L. Reggiani, The Monte Carlo method for the solution of charge transport in semiconductors with applications to covalent materials, Rev. Mod. Phys., 55 (1983), 645-705.
[21]	J. Keilson and J. E. Storer, On Brownian motion, Boltzmann's equation, and the Fokker-Planck equation, Quart. Appl. Math., 10 (1952), 243-253. doi: 10.1090/qam/50216.
[22]	D. S. Kosov, Telegraph noise in Markovian master equation for electron transport through molecular junctions, J. Chem. Phys., 148 (2018).
[23]	K. Lakshmi, R. Parvathy, S. Soumya and K. Soman, Image denoising solutions using heat diffusion equation, In 2012 International Conference on Power, Signals, Controls and Computation, IEEE, (2012), 1-5.
[24]	K. Latrach and B. Lods, Spectral analysis of transport equations with bounce-back boundary conditions, Math. Methods Appl. Sci., 32 (2009), 1325-1344. doi: 10.1002/mma.1088.
[25]	Q. Li, L. Wang and Y. Yang, Monte Carlo gradient in optimization constrained by radiative transport equation, SIAM J. Numer. Anal., 61 (2023), 2744-2774. doi: 10.1137/22M1524515.
[26]	R. D. Skeel, G. Zhang and T. Schlick, A family of symplectic integrators: Stability, accuracy, and molecular dynamics applications, SIAM J. Sci. Comput., 18 (1997), 203-222. doi: 10.1137/S1064827595282350.
[27]	J. Speyer and R. Evans, A second variational theory for optimal periodic processes, IEEE Trans. Automat. Contr., 29 (1984), 138-148. doi: 10.1109/TAC.1984.1103482.
[28]	M. L. Strekalov, Population relaxation of highly rotationally excited molecules at collisions, Chem. Phys. Lett., 548 (2012), 7-11.
[29]	H. Tran, J.-M. Hartmann, F. Chaussard and M. Gupta, An isolated line-shape model based on the Keilson-Storer function for velocity changes. ⅱ. molecular dynamics simulations and the q(1) lines for pure h2, J. Chem. Phys., 131 (2009), 154303.
[30]	C. V. M. van der Mee, Trace theorems and kinetic equations for non-divergence-free external forces, Appl. Anal., 41 (1991), 89-110. doi: 10.1080/00036819108840017.
[31]	L. Verlet, Computer "experiments" on classical fluids. Ⅰ. Thermodynamical properties of Lennard-Jones molecules, Physical Review, 159 (1967), 98.
[32]	Y. Yang, D. Silantyev and R. Caflisch, Adjoint DSMC for nonlinear spatially-homogeneous Boltzmann equation with a general collision model, J. Comput. Phys., 488 (2023), Paper No. 112247. doi: 10.1016/j.jcp.2023.112247.