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<p>In this manuscript, we investigate the global well-posedness of the Hamilton-Jacobi equation for linear-quadratic (LQ) mean field control problems with a common noise, along with the corresponding <i>N</i>-particle systems. The mean field control problems considered are not standard LQ mean field control problems in the sense that their dependence on the mean field terms can be non-convex. The key idea to solving our problem is to utilize the common noise. In contrast to the LQ mean field games master equations, the Hamilton-Jacobi equation for the LQ mean field control problems is inherently an infinite-dimensional partial differential equation which we can show that it cannot be reduced to finite-dimensional one. We then globally solve the Hamilton-Jacobi equation for <i>N</i>-particle systems. As byproducts, we derive the optimal quantitative convergence results from the <i>N</i>-particle systems to the mean field control problems and the propagation of chaos property for the related optimal trajectories. This paper extends the results in [<span>M. Li, C. Mou, Z. Wu and C. Zhou</span>, <i>Trans. Amer. Math. Soc.</i>, 376(06) (2023), pp.&#xa0;4105–4143] to the LQ mean field control problems.</p>

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Global Well-posedness of Hamilton-Jacobi Equations for Linear-Quadratic Mean Field Control Problems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年6月5日

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  • Mengzhen Li,
  • Chenchen Mou,
  • Zhen Wu,
  • Chao Zhou

摘要

In this manuscript, we investigate the global well-posedness of the Hamilton-Jacobi equation for linear-quadratic (LQ) mean field control problems with a common noise, along with the corresponding N-particle systems. The mean field control problems considered are not standard LQ mean field control problems in the sense that their dependence on the mean field terms can be non-convex. The key idea to solving our problem is to utilize the common noise. In contrast to the LQ mean field games master equations, the Hamilton-Jacobi equation for the LQ mean field control problems is inherently an infinite-dimensional partial differential equation which we can show that it cannot be reduced to finite-dimensional one. We then globally solve the Hamilton-Jacobi equation for N-particle systems. As byproducts, we derive the optimal quantitative convergence results from the N-particle systems to the mean field control problems and the propagation of chaos property for the related optimal trajectories. This paper extends the results in [M. Li, C. Mou, Z. Wu and C. Zhou, Trans. Amer. Math. Soc., 376(06) (2023), pp. 4105–4143] to the LQ mean field control problems.