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Fast and Asymptotic Steering to a Steady State for Networks Flows

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Book cover Geometric Science of Information (GSI 2021)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 12829))

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Abstract

We study the problem of optimally steering a network flow to a desired steady state, such as the Boltzmann distribution with a lower temperature, both in finite time and asymptotically. In the infinite horizon case, the problem is formulated as constrained minimization of the relative entropy rate. In such a case, we find that, if the prior is reversible, so is the solution.

Supported in part by the NSF under grant 1807664, 1839441, 1901599, 1942523, the AFOSR under grants FA9550-20-1-0029, and by the University of Padova Research Project CPDA 140897.

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Notes

  1. 1.

    \(x=(x_0,x_1,\ldots ,x_N)\) is feasible if \((x_i,x_{i+1})\in \mathcal E, \forall i\).

  2. 2.

    Relative entropy (divergence, Kullback-Leibler index) is defined by

    $$\begin{aligned} {\mathbb D}(\mathfrak P\Vert \mathfrak M):=\left\{ \begin{array}{ll} \sum _{x}\mathfrak P(x)\log \frac{\mathfrak P(x)}{\mathfrak M(x)}, &{} \mathrm{Supp}(\mathfrak P)\subseteq \mathrm{Supp}(\mathfrak M),\\ +\infty , &{} \mathrm{Supp}(\mathfrak P)\not \subseteq \mathrm{Supp}(\mathfrak M),\end{array}\right. \end{aligned}$$

    Here, by definition, \(0\cdot \log 0=0\). The value of \({\mathbb D}(\mathfrak P\Vert \mathfrak M)\) may turn out to be negative due to the different total masses in the case when \({\mathfrak M}\) is not a probability measure. The optimization problem, however, poses no challenge as the relative entropy is (jointly) convex over this larger domain and bounded below.

  3. 3.

    Please see the Introduction for the historical justification of this name.

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Authors and Affiliations

  1. School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Yongxin Chen

  2. Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, 92697, USA

    Tryphon Georgiou

  3. Department of Mathematics “Tullio Levi-Civita”, University of Padova, 35121, Padova, Italy

    Michele Pavon

Authors
  1. Yongxin Chen
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  2. Tryphon Georgiou
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  3. Michele Pavon
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Corresponding author

Correspondence to Tryphon Georgiou .

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Editors and Affiliations

  1. Sony Computer Science Laboratories Inc., Tokyo, Japan

    Frank Nielsen

  2. THALES Land & Air Systems, Meudon, France

    Frédéric Barbaresco

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Chen, Y., Georgiou, T., Pavon, M. (2021). Fast and Asymptotic Steering to a Steady State for Networks Flows. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_92

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  • DOI: https://doi.org/10.1007/978-3-030-80209-7_92

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