This paper is concerned with the open-loop time-consistent solution of time-inconsistent mean-field stochastic linear-quadratic (LQ) optimal control. Different from standard stochastic linear-quadratic problems, both the system matrices and the weighting matrices are depending on the initial times, and the conditional expectations of the control and state enter quadratically into the cost functional. Such features will ruin Bellman's principle of optimality and result in the time inconsistency of optimal control. Based on the dynamical nature of the systems involved, a kind of open-loop time-consistent equilibrium control is investigated in this paper. It is shown that the existence of open-loop equilibrium control for a fixed initial pair is equivalent to the solvability of a set of forward–backward stochastic difference equations with stationary condition and convexity condition. By decoupling the forward–backward stochastic difference equations, necessary and sufficient conditions in terms of linear difference equations and generalized difference Riccati equations are given for the existence of open-loop equilibrium control for a fixed initial pair. Moreover, the existence of open-loop time-consistent equilibrium controls for all the initial pairs is shown to be equivalent to the solvability of a set of coupled constrained generalized difference Riccati equations and two sets of constrained linear difference equations.

中文翻译：

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时间不一致平均场随机 LQ 问题：开环时间一致控制

本文关注的是时间不一致平均场随机线性二次 (LQ) 最优控制的开环时间一致解。与标准随机线性二次问题不同，系统矩阵和加权矩阵都取决于初始时间，控制和状态的条件期望二次进入成本函数。这样的特征会破坏贝尔曼的最优性原理，导致最优控制的时间不一致。基于所涉及系统的动力学特性，本文研究了一种开环时间一致平衡控制。结果表明，对于固定初始对的开环平衡控制的存在等价于一组具有平稳条件和凸性条件的前向后向随机差分方程的可解性。通过解耦前向-后向随机差分方程，给出了线性差分方程和广义差分Riccati方程对于固定初始对存在开环平衡控制的充要条件。此外，所有初始对的开环时间一致平衡控制的存在被证明等效于一组耦合约束广义差分 Riccati 方程和两组约束线性差分方程的可解性。通过解耦前向-后向随机差分方程，给出了线性差分方程和广义差分Riccati方程对于固定初始对存在开环平衡控制的充要条件。此外，所有初始对的开环时间一致平衡控制的存在被证明等效于一组耦合约束广义差分 Riccati 方程和两组约束线性差分方程的可解性。通过解耦前向-后向随机差分方程，给出了线性差分方程和广义差分Riccati方程对于固定初始对存在开环平衡控制的充要条件。此外，所有初始对的开环时间一致平衡控制的存在被证明等效于一组耦合约束广义差分 Riccati 方程和两组约束线性差分方程的可解性。