近年來，平均場隨機系統(Mean Field Stochastic System)已成為熱門研究領域，被廣泛地應用於工程、物理學、隨機控制、博局論與計量金融等領域。平均場隨機動態系統與一般隨機動態系統最大的差別在於，其動態方程中出現了系統狀態的平均項，而這平均項的出現導致設計平均場隨機動態系統控制器要難上很多。近年的研究，主要在於線性平均場隨機動態系統的成本最佳化設計與強健控制，但目前多只有探討外部雜訊與Wiener process對線性平均場系統造成的影響(即狀態軌跡連續的線性平均場系統)。實際上，多數隨機系統為非線性系統且同時包含Wiener process及Poisson jump process的狀態相依擾動，因此有必要徹底研究Poisson jump process的線性與非線性隨機平均場系統。在實際應用上，希望能兼顧控制器的成本最佳化與強健性，故需導入多目標控制器設計。本論文主要目標在於提出有效率的多目標控制器設計應用在Poisson jump process的線性與非線性隨機平均場系統上，並同時探討Poisson jump process的非線性隨機平均場系統的穩定度、最佳化成本調控設計、強健控制器設計多目標控制器並將其應用在工程、物理及金融系統上。本計畫探討有Wiener process及Poisson process下的非線性隨機平均場的多目標控制設計，並用金融系統作實例。

In recent years, many researchers have paid their attention to mean field stochastic system. Now, the control design problem of the mean field stochastic system has become an active research field and can be employed in the control designs of engineering, physical, game theory, stochastic and quantitative finance systems. The main difference between a mean field stochastic control design problem and a classical stochastic control design problem is that the mean term ( ) appears in mean field system. For the mean field stochastic control design problems, its controller is more difficult to design than the classic one because of the mean term in their stochastic dynamic systems. In the last decade, most researchers have focused on the optimal cost control and robust control of linear mean field stochastic system with continuous internal disturbance. Only a few studies discussed the linear mean field stochastic system with Poisson jump diffusions. However, most stochastic systems are nonlinear. Moreover, many practical systems not only contain the continuous state dependent intrinsic fluctuation caused by Wiener process but also contain the discontinuous state dependent intrinsic jumping process caused by Poisson process. Further, these mean field stochastic systems are also computed by the external disturbance. It is necessary to do an advance research on the robust and optimal design of nonlinear mean field stochastic system with Poisson jump diffusions for the practical applications. Because engineers always expect the designed controller to be not only optimal but also robust, the conventional single objective optimal cost control design or robust control design may not enough to handle the practical nonlinear stochastic system, respectively. Thus, it is more appealing to propose the multi objective control design to simultaneously achieve the optimal control and robust control for the mean field stochastic system, simultaneously. Our aim in this project is to design a multi-objective controller for the nonlinear mean field stochastic jump diffusion system and then apply the proposed multi-objective controller for the nonlinear mean field stochastic jump diffusion system to engineering, physical and quantitative finance systems.

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