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The turnpike property for high-dimensional interacting agent systems in discrete time

Martin Gugat, Michael Herty, Jiehong Liu, Chiara Segala

TL;DR

The paper extends the interior turnpike property from continuous-time multi-agent optimal control to the time-discrete setting using an explicit Euler discretization. By establishing a discrete dynamic programming principle, a strict dissipativity inequality with $\alpha(x)=\frac{\gamma}{2N}x^2$, and a cheap-control strategy, it proves that the discrete-time problem exhibits a turning-point behavior with interior decay that is independent of the number of agents $N$ and, crucially, of the time-step $h$. The analysis shows uniformity as $h\to 0^+$ and is complemented by numerical experiments across various discretizations, agent counts, and even non-differentiable interaction kernels, confirming the theoretical predictions. This work supports the practical use of discrete-time control for large-scale agent systems and informs efficient computational strategies via the static (turnpike) solution for long horizons.

Abstract

We investigate the interior turnpike phenomenon for discrete-time multi-agent optimal control problems. While for continuous systems the turnpike property has been established, we focus here on first-order discretizations of such systems. It is shown that the resulting time-discrete system inherits the turnpike property with estimates of the same type as in the continuous case. In particular, we prove that the discrete time optimal control problem is strictly dissipative and the cheap control assumption holds.

The turnpike property for high-dimensional interacting agent systems in discrete time

TL;DR

The paper extends the interior turnpike property from continuous-time multi-agent optimal control to the time-discrete setting using an explicit Euler discretization. By establishing a discrete dynamic programming principle, a strict dissipativity inequality with , and a cheap-control strategy, it proves that the discrete-time problem exhibits a turning-point behavior with interior decay that is independent of the number of agents and, crucially, of the time-step . The analysis shows uniformity as and is complemented by numerical experiments across various discretizations, agent counts, and even non-differentiable interaction kernels, confirming the theoretical predictions. This work supports the practical use of discrete-time control for large-scale agent systems and informs efficient computational strategies via the static (turnpike) solution for long horizons.

Abstract

We investigate the interior turnpike phenomenon for discrete-time multi-agent optimal control problems. While for continuous systems the turnpike property has been established, we focus here on first-order discretizations of such systems. It is shown that the resulting time-discrete system inherits the turnpike property with estimates of the same type as in the continuous case. In particular, we prove that the discrete time optimal control problem is strictly dissipative and the cheap control assumption holds.
Paper Structure (15 sections, 5 theorems, 45 equations, 9 figures)

This paper contains 15 sections, 5 theorems, 45 equations, 9 figures.

Table of Contents

  1. Keywords:
  2. AMS Classification:
  3. Introduction
  4. The discrete-time optimal control problem
  5. The turnpike property with interior decay
  6. The strict dissipativity property
  7. The cheap control condition
  8. Proof of Theorem \ref{['thm1']}
  9. Uniformity of the turnpike property for $h\rightarrow 0^+$
  10. Numerical experiments
  11. Test 1: Trajectories and controls, the turnpike phenomenon.
  12. Test 2: Comparison for different time-steps.
  13. Test 3: Comparison for increasing numbers of agents.
  14. Test 4: Non-differentiable interaction kernel.
  15. Conclusion

Key Result

Lemma 2.1

Let $(\hat{\psi}, \hat{u})$ be the optimal state-control pair for the optimal control problem $Q(t_{0},t_{L},h,N,\psi^{0})$ for some $t_0<t_L < T.$ Then, $\hat{\psi}\mid _{(a,L)}=(\hat{\psi}^{a}, \hat{\psi}^{a+1},\dots, \hat{\psi}^{L})$ and $\hat{u}\mid _{(a,L)}=(\hat{u}^{a}, \hat{u}^{a+1},\dots, \h

Figures (9)

  • Figure 1: Test 1. Trajectories evolution in time for $N=100$ agents without control. The uncontrolled dynamics converge to its equilibrium point.
  • Figure 2: Test 1. From left to right, evolution of states (top) and controls (bottom) for $N=100$ agents which are steered by cheap control with $\beta=1,3,8$ and optimal control respectively. The larger the value of $\beta$, the faster the agents reach the desired state.
  • Figure 3: Test 1. On the top, the Lyapunov functions on a logarithmic scale (left), and the running cost over time (right). On the bottom, the turnpike property with interior decay, i.e. the gap between the optimal and static states of all particles $\sum_{k=1}^{N}\psi^{i}_{k} - \sum_{k=1}^{N}\bar{\psi}$ (left) and the gap between optimal and static control $\sum_{k=1}^{N}u^{i}_{k}$ (right) over time.
  • Figure 4: Test 2. From left to right, trajectories of state (top) and control (bottom) for discretization levels $h=0.1, 0.01, 0.001$.
  • Figure 5: Test 2. The exponential decay of $L_N$ for different discretization levels $h=0.1, 0.01, 0.001$.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Lemma 2.1
  • proof
  • Definition 2.1
  • Theorem 2.2
  • Definition 3.1
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.2
  • proof
  • ...and 2 more