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Sparse control in microscopic and mean-field leader-follower models

Melanie Harms, Michael Herty, Chiara Segala, Eva Zerz

Abstract

This work investigates the decay properties of Lyapunov functions in leader-follower systems seen as a sparse control framework. Starting with a microscopic representation, we establish conditions under which the total Lyapunov function, encompassing both leaders and followers, exhibits exponential decay. The analysis is extended to a hybrid setting combining a mean-field description for followers and a microscopic model for leaders. We identify sufficient conditions on control gain and interaction strengths that guarantee stabilization of the linear system towards a target state. The results highlight the influence of sparse control and interaction parameters in achieving coordinated behavior in multi-agent systems.

Sparse control in microscopic and mean-field leader-follower models

Abstract

This work investigates the decay properties of Lyapunov functions in leader-follower systems seen as a sparse control framework. Starting with a microscopic representation, we establish conditions under which the total Lyapunov function, encompassing both leaders and followers, exhibits exponential decay. The analysis is extended to a hybrid setting combining a mean-field description for followers and a microscopic model for leaders. We identify sufficient conditions on control gain and interaction strengths that guarantee stabilization of the linear system towards a target state. The results highlight the influence of sparse control and interaction parameters in achieving coordinated behavior in multi-agent systems.

Paper Structure

This paper contains 12 sections, 4 theorems, 95 equations, 5 figures.

Table of Contents

  1. Keywords:
  2. AMS Classification:
  3. Introduction
  4. Multi-agent system with control action
  5. Eigenvalue analysis of the linearized system
  6. Mean-field limit
  7. Leader-follower model
  8. Lyapunov stability analysis
  9. Full control Lyapunov decay
  10. Sparse control Lyapunov decay
  11. Numerical tests
  12. Conclusion

Key Result

Proposition 4.2

If Assumption ass:P_non_negative holds and the control eq:u_feedback with control parameter $k<0$ is applied uniformly to all agents, then the Lyapunov function $\mathcal{L}(t)$ satisfies the following inequality: for all $t$ with $x(t)\neq x_c$.

Figures (5)

  • Figure 1: Evolution of the uncontrolled system. Left column: microscopic trajectories (top) and mean-field density (bottom). Right column: Lyapunov function decay in microscopic and mean-field settings.
  • Figure 2: Effect of full control on the system. Left column: microscopic trajectories (top) and mean-field density (bottom). Right column: Lyapunov function decays.
  • Figure 3: Sparse control dynamics. Left column: mean-field dynamics for the non-linear (top) and linear (bottom) case. Right column: Lyapunov function evolution over time.
  • Figure 4: Leader-follower system. Left column: microscopic trajectories (top) and mean-field density (bottom). Right column: Lyapunov function decays.
  • Figure 5: Lyapunov function decay for different values of $\omega^L$ under the leader-follower scenario.

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Proposition 4.2: Lyapunov function decay: Microscopic full control
  • proof
  • Proposition 4.3: Lyapunov function decay: Mean-field full control
  • proof
  • Proposition 4.4: Lyapunov Function Decay: Microscopic Sparse Control
  • proof
  • Proposition 4.5: Lyapunov Function Decay: Mean-field Sparse Control
  • proof