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Turnpike property for hierarchical optimal control problems: from particle systems to hydrodynamic equations

Michael Herty, Yizhou Zhou

TL;DR

This work analyzes exponential turnpike properties across a hierarchy of controlled dynamics: an $N$-particle system, its mean-field limit, and macroscopic hydrodynamic closures. By constructing feedback controls and proving cheap-control properties at each level, it derives Gronwall-based bounds that yield exponential convergence toward steady states as the horizon grows. The main contributions are the uniform-in-$N$ turnpike estimates for the particle system, their rigorous mean-field passage, and the analogous results for two hydrodynamic closures (Type I and Type II Euler equations) with both pressure-less and full-energy formulations. The results provide a unified framework linking microscopic, kinetic, and macroscopic optimal-control descriptions, with potential implications for scalable stabilization and control of multi-scale collective dynamics.

Abstract

This work is concerned with a hierarchical framework of optimal control problems connecting interacting particle systems, the mean field limit equations, and associated hydrodynamic models. By assuming the existence of solutions, we establish the exponential turnpike property for each level of the hierarchy, showing that optimal trajectories remain close to the associated steady states over long time horizons. The results demonstrate that the exponential turnpike behavior persists consistently across scales, providing a unified connection between microscopic, kinetic, and macroscopic optimal control frameworks.

Turnpike property for hierarchical optimal control problems: from particle systems to hydrodynamic equations

TL;DR

This work analyzes exponential turnpike properties across a hierarchy of controlled dynamics: an -particle system, its mean-field limit, and macroscopic hydrodynamic closures. By constructing feedback controls and proving cheap-control properties at each level, it derives Gronwall-based bounds that yield exponential convergence toward steady states as the horizon grows. The main contributions are the uniform-in- turnpike estimates for the particle system, their rigorous mean-field passage, and the analogous results for two hydrodynamic closures (Type I and Type II Euler equations) with both pressure-less and full-energy formulations. The results provide a unified framework linking microscopic, kinetic, and macroscopic optimal-control descriptions, with potential implications for scalable stabilization and control of multi-scale collective dynamics.

Abstract

This work is concerned with a hierarchical framework of optimal control problems connecting interacting particle systems, the mean field limit equations, and associated hydrodynamic models. By assuming the existence of solutions, we establish the exponential turnpike property for each level of the hierarchy, showing that optimal trajectories remain close to the associated steady states over long time horizons. The results demonstrate that the exponential turnpike behavior persists consistently across scales, providing a unified connection between microscopic, kinetic, and macroscopic optimal control frameworks.
Paper Structure (20 sections, 12 theorems, 118 equations, 5 figures)

This paper contains 20 sections, 12 theorems, 118 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Hierarchy of control problems
  3. Optimal control problem for the particle system
  4. Optimal control problem for the mean field equation
  5. Optimal control problem for the hydrodynamic equation
  6. Type I:
  7. Type II:
  8. Feedback control and cheap control property
  9. Particle systems
  10. Mean field level
  11. Hydrodynamic level
  12. Pressure-less Euler equation
  13. The full Euler equation
  14. Exponential turnpike property
  15. Proof of the theorem for the particle system
  16. ...and 5 more sections

Key Result

Theorem 2.1

Assume that the initial data $\mu_0\in P(\mathbb{R}^D)$ in mean-field-pb is compactly supported, i.e., there exists $R > 0$ such that $\text{supp}~ \mu_0 \subset B(0, R)\subset \mathbb{R}^{2D}$. Moreover, the empirical measure $\mu_N(0,x,v)$ converges to $\mu_0$ in $\mathcal{W}_1$ distance. Then, th for any $\phi\in C_0^{\infty}([0,T]\times \mathbb{R}^{2D})$. The optimal solution satisfies unifor

Figures (5)

  • Figure 1: Numerical results for the particle system: (left) position $x_k$ of each particle; (middle) velocity $v_k$ of each particle; (right) the decay of the cost, represented in the logarithmic regime.
  • Figure 2: Numerical results for the pressure-less hydrodynamic equation: (left) the initial data at $t=0$; (middle) the solution at $t=0.5$; (right) the solution at $t=4$.
  • Figure 3: The pressure-less hydrodynamic equation: the decay of the cost function $\rho|u-\bar{v}|^2+\gamma|f_H|^2$ with feedback law, represented in the logarithmic regime.
  • Figure 4: Numerical results for the Euler equation: (left) the initial data at $t=0$; (middle) the solution at $t=0.5$; (right) the solution at $t=4$.
  • Figure 5: The Euler equation: the decay of the cost function $\rho|u-\bar{v}|^2+2\rho e+\gamma G(F_1,F_2)$ with feedback law, represented in the logarithmic regime.

Theorems & Definitions (25)

  • Theorem 2.1
  • Remark 2.1
  • Lemma 3.1: Cheap control
  • proof
  • Remark 3.1
  • Lemma 3.2: Cheap control
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 15 more