Dynamic optimization problems for mean-field stochastic large-population systems

IF 1.3 3区数学 Q4 AUTOMATION & CONTROL SYSTEMS

Esaim-Control Optimisation and Calculus of Variations Pub Date : 2022-06-16 DOI:10.1051/cocv/2022044

Min Li, Na Li, Zhanghua Wu

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引用次数: 2

Abstract

This paper considers dynamic optimization problems for a class of control average mean-field stochastic large-population systems. For each agent, the state system is governed by a linear mean-field stochastic differential equation (MF-SDE) with individual noise and common noise, and the weight coefficients in the cost functional can be indefinite. The decentralized optimal strategies are characterized by stochastic Hamiltonian system, which turns out to be an algebra equation and a mean-field forward-backward stochastic differential equation (MF-FBSDE). Applying the decoupling method, the feedback representation of decentralized optimal strategies is further obtained through two Riccati equations. The solvability of stochastic Hamiltonian system and Riccati equations under indefinite condition is also derived. The explicit structure of the control average limit and the related mean-field Nash certainty equivalence (NCE) equation systems are also discussed by some separation techniques. Moreover, the decentralized optimal strategies are proved to satisfy the approximate Nash equilibrium property. The good performance of the proposed theoretical results is illustrated by a practical example from the engineering field.

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平均场随机大种群系统的动态优化问题

研究了一类控制平均平均场随机大种群系统的动态优化问题。对于每个智能体，状态系统由一个具有个体噪声和共同噪声的线性平均场随机微分方程(MF-SDE)控制，并且代价函数中的权重系数可以是不定的。分散最优策略的特征是随机哈密顿系统，即代数方程和平均场正反向随机微分方程(MF-FBSDE)。应用解耦方法，进一步通过两个Riccati方程得到分散最优策略的反馈表示。导出了随机哈密顿系统和Riccati方程在不定条件下的可解性。利用分离技术讨论了控制平均极限的显式结构和相关的平均场纳什确定性等价方程组。此外，还证明了分散最优策略满足近似纳什均衡性质。通过工程实例验证了所提理论结果的正确性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Esaim-Control Optimisation and Calculus of Variations Mathematics-Computational Mathematics

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7.10%

发文量

期刊介绍： ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations. Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines. Targeted topics include: in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory; in optimisation: mathematical programming, large scale systems, stochastic optimisation, combinatorial optimisation, shape optimisation, convex or nonsmooth optimisation, inverse problems, interior point methods, duality methods, numerical methods, convergence and complexity, global optimisation, optimisation and dynamical systems, optimal transport, machine learning, image or signal analysis; in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.