二阶平均场薛定谔桥与部分状态信息
Second-Order Mean-Field Schrödinger Bridges with Partial State Information
- Iowa State University（爱荷华州立大学）
- Scuola Superiore Meridionale（南方高等学院）
- Georgia Institute of Technology（佐治亚理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对部分状态信息下的惯性相互作用智能体系统，提出二阶平均场薛定谔桥新表述，推导最优性系统并开发嵌套不动点算法求解。
AI中文摘要：
受控制系统中决策仅基于状态的部分信息这一场景的启发，我们针对惯性相互作用智能体系统提出了一种薛定谔桥问题的新表述。在我们的设置中，智能体集合被引导以匹配集体配置的部分规格。我们的分析涉及一个平均场框架，其中集体动力学由具有非局部相互作用的非线性Vlasov--Fokker--Planck积分偏微分方程描述。我们假设在有限时间范围的端点仅指定位置上的分布，代表总体种群快照，而相应的速度自由度保持未指定。我们推导了相关的最优性系统，将最优受控演化识别为通过时间上的前向和后向传播，由初始和终端规格共同告知。我们还表明，部分端点规格在最优控制和一致演化上带来了不同的条件。随后，为了计算最优性系统的解，我们开发了一种嵌套不动点算法，该算法受经典Fortet--Sinkhorn迭代的启发，但扩展以处理由非局部相互作用和时间耦合引起的非线性。
英文摘要：
Motivated by scenarios in control systems where decision making is informed by only partial information about the states, we introduce a novel formulation of the Schrödinger bridge problem for inertial interacting-agent systems. Within our setting, the agent ensemble is steered to match partial specifications of the collective configuration. Our analysis pertains to a mean-field framework in which the collective dynamics are described by a nonlinear Vlasov--Fokker--Planck integro-PDE with nonlocal interactions. We assume that only distributions over positions are specified at the endpoints of a finite time horizon, representing aggregate population snapshots, while the corresponding velocity degrees of freedom remain unspecified. We derive the associated optimality system, identifying the optimal controlled evolution as jointly informed by the initial and terminal specifications through forward and backward propagation in time. We also show that the partial endpoint specifications bring about distinct conditions on the optimal control and the consistent evolution. Subsequently, to compute solutions to the optimality system, we develop a nested fixed-point algorithm inspired by the classical Fortet--Sinkhorn iteration, but extended to handle the nonlinearities arising from nonlocal interactions and temporal coupling.
