AI 中文总结
研究线性二次图上均值场控制问题的渐近行为和大道性质，通过Riccati方程组等表征最优控制和状态轨迹，在特定条件下建立遍历控制问题可解性及收敛估计，证明最优对的指数大道性质和时间平均价值函数收敛。
AI 中文摘要
我们研究线性二次设定下的图上均值场控制（GMFC）问题的渐近行为和大道性质。考虑了有限时域GMFC问题及其相关的遍历问题，其受控动力学由具有异质相互作用的图上均值场随机微分方程控制。两个问题的最优控制和状态轨迹由Riccati方程组以及合适希尔伯特空间上的广义微分和代数方程组表征。在图上诱导算子的可镇定条件和适当的正性假设下，建立了遍历控制问题的唯一可解性，并导出了有限时域系统到其平稳极限的指数收敛估计。由此，建立了最优对的指数大道性质，并证明了有限时域GMFC问题的时间平均价值函数的收敛性。
英文摘要
We investigate the asymptotic behavior and turnpike properties of graphon mean field control (GMFC) problems in the linear-quadratic setting. We consider both a finite-horizon GMFC problem and its associated ergodic counterpart, in which the controlled dynamics are governed by a graphon mean field stochastic differential equation with heterogeneous interactions. The optimal controls and state trajectories for both problems are characterized by systems of Riccati equations together with systems of generalized differential and algebraic equations on suitable Hilbert spaces. Under a stabilizability condition and appropriate positivity assumptions on the graphon-induced operators, we establish the unique solvability of the ergodic control problem and derive exponential convergence estimates for the finite-horizon system to its stationary limit. As a consequence, we establish an exponential turnpike property for the optimal pair and prove the convergence of the time-averaged value function for the finite-horizon GMFC problem.
Comments44 pages
