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Risk-Aware Finite-Horizon Social Optimal Control of Mean-Field Coupled Linear-Quadratic Subsystems

Dhairya Patel, Margaret Chapman

TL;DR

This work addresses risk-aware social optimal control for mean-field coupled linear-quadratic subsystems on a finite horizon by incorporating predictive variance into the cost to penalize state-energy variability. The authors reformulate the problem centrally using pseudo-block diagonal matrices and Kronecker-based maps, which enables translating a standard uncoupled LQR solution into a mean-field coupled controller without rederiving from scratch. They establish a rigorous matrix-operator framework that decouples mean-field and subsystem contributions, derive the optimal mean-field coupled control ${u_t^i}^* = K_t x_t^i + (ar{K}_t-K_t)ar{x}_t + f_t$, and show how the centralized Riccati-like quantities decompose via $ ilde{S}_t=oldsymbol{a_k}(S_t,ar{S}_t)$. Numerical results illustrate that larger risk-weights reduce both average and worst-case state energies and their variability, at the cost of higher average control effort, highlighting a controllable trade-off between performance and robustness. The approach provides a scalable pathway to extend classical LQR results to mean-field settings and suggests avenues for broader risk formulations and partially observable extensions.

Abstract

We formulate and solve an optimal control problem with cooperative, mean-field coupled linear-quadratic subsystems and additional risk-aware costs depending on the covariance and skew of the disturbance. This problem quantifies the variability of the subsystem state energy rather than merely its expectation. In contrast to related work, we develop an alternative approach that illuminates a family of matrices with many analytical properties, which are useful for effectively extracting the mean-field coupled solution from a standard LQR solution.

Risk-Aware Finite-Horizon Social Optimal Control of Mean-Field Coupled Linear-Quadratic Subsystems

TL;DR

This work addresses risk-aware social optimal control for mean-field coupled linear-quadratic subsystems on a finite horizon by incorporating predictive variance into the cost to penalize state-energy variability. The authors reformulate the problem centrally using pseudo-block diagonal matrices and Kronecker-based maps, which enables translating a standard uncoupled LQR solution into a mean-field coupled controller without rederiving from scratch. They establish a rigorous matrix-operator framework that decouples mean-field and subsystem contributions, derive the optimal mean-field coupled control , and show how the centralized Riccati-like quantities decompose via . Numerical results illustrate that larger risk-weights reduce both average and worst-case state energies and their variability, at the cost of higher average control effort, highlighting a controllable trade-off between performance and robustness. The approach provides a scalable pathway to extend classical LQR results to mean-field settings and suggests avenues for broader risk formulations and partially observable extensions.

Abstract

We formulate and solve an optimal control problem with cooperative, mean-field coupled linear-quadratic subsystems and additional risk-aware costs depending on the covariance and skew of the disturbance. This problem quantifies the variability of the subsystem state energy rather than merely its expectation. In contrast to related work, we develop an alternative approach that illuminates a family of matrices with many analytical properties, which are useful for effectively extracting the mean-field coupled solution from a standard LQR solution.
Paper Structure (7 sections, 10 theorems, 24 equations, 1 figure)

This paper contains 7 sections, 10 theorems, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Formulation and Notation
  3. Centralized Reformulation
  4. Pseudo-Block Diagonal Matrices
  5. Optimal Control
  6. Numerical Example
  7. Conclusion

Key Result

Proposition 1

eq:mean_field_dynamics can be rewritten as where $\tilde{A}_t \triangleq I_k \otimes A_t + E_k \otimes C_t$ and $\tilde{B}_t \triangleq I_k \otimes B_t$.

Figures (1)

  • Figure 1: Plots of \ref{['fig:avg_state_energy_vs_t']} average state energy $c_t^{x, \mathrm{avg}} \triangleq \frac{1}{k} \sum_{i \in \mathbb{I}} (x_t^i)^\top Q_t x_t^i$ vs. $t$, \ref{['fig:max_state_energy_vs_t']} maximum state energy $c_t^{x, \mathrm{max}} \triangleq \max_{i \in \mathbb{I}} (x_t^i)^\top Q_t x_t^i$ vs. $t$, \ref{['fig:state_energy_vs_lambda']} time average of state energy statistics, $c_t^{x, \mathrm{avg}}$ and $c_t^{x, \mathrm{max}}$, vs. $\lambda$, and \ref{['fig:control_effort_vs_lambda']} time average of control effort statistics, $c_t^{u, \mathrm{avg}} \triangleq \frac{1}{k} \sum_{i \in \mathbb{I}} (u_t^i)^\top R_t u_t^i$ and $c_t^{u, \mathrm{max}} \triangleq \max_{i \in \mathbb{I}} (u_t^i)^\top R_t u_t^i$, vs. $\lambda$. The horizontal axis labels are \ref{['fig:avg_state_energy_vs_t']} time $t$, \ref{['fig:max_state_energy_vs_t']} time $t$, \ref{['fig:state_energy_vs_lambda']}$\lambda$, and \ref{['fig:control_effort_vs_lambda']}$\lambda$. The bold lines and shaded regions indicate the empirical mean values and observed credible intervals, respectively, of the plotted quantities among the $10^4$ simulations.

Theorems & Definitions (20)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Lemma 1
  • proof
  • ...and 10 more