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Fokker--Planck Dynamics on Star Graphs with Variable Drift: Well-Posedness, Adjoint Analysis, and Numerical Approximation

Ritu Kumari, Cyrille Kenne, Landry Djomegne, Mani Mehra

TL;DR

The paper addresses optimal control of Fokker--Planck dynamics on a star-graph network, with the control entering bilinearly through the drift. It develops a rigorous well-posedness theory for the state equation, proves existence of an optimal control, and derives the adjoint system together with first-order optimality conditions. A wavelet-based collocation scheme using shifted Legendre wavelets is proposed to numerically solve the resulting optimality system, and two numerical experiments demonstrate high accuracy and convergence. The results advance analytical and computational understanding of controlled stochastic transport on networked domains and provide a practical numerical tool for network-structured FP control problems.

Abstract

Stochastic transport processes on networked domains (modelled on metric graphs) arise in a variety of applications where diffusion and drift mechanisms interact with an underlying graph structure. The Fokker--Planck equation provides a natural framework for describing the evolution of probability densities associated with such dynamics. While Fokker--Planck equations on metric graphs have been studied from an analytical viewpoint, their optimal control remains largely unexplored, particularly in settings where the control acts through the drift term. In this paper, we investigate an optimal control problem governed by the Fokker--Planck equation on a star graph, with a bilinear control appearing in the drift. We establish the well-posedness of the state equation and prove the existence of at least one optimal control. The associated adjoint system is derived, and first-order necessary optimality conditions are formulated. A wavelet-based numerical scheme is proposed to approximate the optimal solution, and its performance is illustrated through representative numerical experiments. These results contribute to the analytical and computational understanding of controlled stochastic dynamics on network-like domains.

Fokker--Planck Dynamics on Star Graphs with Variable Drift: Well-Posedness, Adjoint Analysis, and Numerical Approximation

TL;DR

The paper addresses optimal control of Fokker--Planck dynamics on a star-graph network, with the control entering bilinearly through the drift. It develops a rigorous well-posedness theory for the state equation, proves existence of an optimal control, and derives the adjoint system together with first-order optimality conditions. A wavelet-based collocation scheme using shifted Legendre wavelets is proposed to numerically solve the resulting optimality system, and two numerical experiments demonstrate high accuracy and convergence. The results advance analytical and computational understanding of controlled stochastic transport on networked domains and provide a practical numerical tool for network-structured FP control problems.

Abstract

Stochastic transport processes on networked domains (modelled on metric graphs) arise in a variety of applications where diffusion and drift mechanisms interact with an underlying graph structure. The Fokker--Planck equation provides a natural framework for describing the evolution of probability densities associated with such dynamics. While Fokker--Planck equations on metric graphs have been studied from an analytical viewpoint, their optimal control remains largely unexplored, particularly in settings where the control acts through the drift term. In this paper, we investigate an optimal control problem governed by the Fokker--Planck equation on a star graph, with a bilinear control appearing in the drift. We establish the well-posedness of the state equation and prove the existence of at least one optimal control. The associated adjoint system is derived, and first-order necessary optimality conditions are formulated. A wavelet-based numerical scheme is proposed to approximate the optimal solution, and its performance is illustrated through representative numerical experiments. These results contribute to the analytical and computational understanding of controlled stochastic dynamics on network-like domains.
Paper Structure (14 sections, 8 theorems, 81 equations, 13 figures, 6 tables)

This paper contains 14 sections, 8 theorems, 81 equations, 13 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Literature review
  3. Problem Formulation
  4. Contribution
  5. Preliminary results
  6. Existence results
  7. Existence of optimal solutions and first-order optimality conditions
  8. Regularity results for the control-to-state operator
  9. The adjoint equation and first-order necessary optimality conditions
  10. Numerical Illustration
  11. Shifted Legendre Wavelets
  12. Numerical Scheme
  13. Numerical Results
  14. Conclusion

Key Result

Theorem 3.2

Let $f\in L^2(0,T;\mathbb{V}^\star)$, $u\in \mathbb{L}^\infty$ and $\rho^{0}\in \mathbb{L}^2$. Then, there exists a unique weak solution $\rho=(\rho_i)_i\in \mathbb{W}(0,T)$ to general in the sense of Definition weakgen. In addition, there exists a constant $C>0$ depending on $T$, $\underbar{D}$, $\

Figures (13)

  • Figure 1: A star graph having $N$ edges.
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 8 more figures

Theorems & Definitions (23)

  • Definition 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • Remark 3.4
  • Theorem 4.1
  • proof
  • Remark 4.2
  • Definition 4.3
  • Lemma 4.4
  • ...and 13 more