在本文中，我們考慮一個多輸入者非線性隨機跳躍擴散系統。假設每個輸入者被視為一個玩家，這個多輸入者非線性隨機跳躍擴散系統可以被看作是一個不互相交換訊息的非合作差分賽局。將隨機多玩家非合作賽局策略問題轉化為特徵值多元Halmition Jacobi不等式(HJIs) 約束下的多目標優化問題。為了方便設計，我們使用Takagi - Sugeno（T-S） 模糊模型來逼近這個非線性系統。因此HJI約束下的多目標優化問題可以轉化為線性矩陣不等式（LMIs）約束下的多目標優化問題。另一方面，為了有效解決非線性隨機跳躍擴散系統中的多玩家非合作博弈策略，還開發了LMIs約束下的多目標進化演算法（MOEA）。採用MOEA來優化所有參與者的目標，並提出多目標最優控制器的設計過程。之後，我們可以獲得非合作的策略，可以有效地為每個玩家實現最好的目標，而犧牲其他目標。最後，為了說明所提出的多目標最優控制設計程序的有效性，提供了金融市場的模擬。

In this paper, we consider a multi-agents nonlinear stochastic jump diffusion system. Assuming that each agent is treated as one player, this multi-agents nonlinear stochastic jump diffusion system can be treated as a non-cooperative differential game without intercommunication with each other. The stochastic multi-player non-cooperative game strategy problem is transformed to eigenvalues multi-tuple Hamilton Jacobi inequalities (HJIs)-constrained multi-objective optimization problem (MOP). For convenience designed, we use Takagi-Sugeno (T-S) fuzzy model to approximate this nonlinear system. So that the HJIs-constrained MOP could be transformed into a linear matrix inequalities (LMIs)-constrained MOP problem. On the other hand, an LMIs-constrained multi-objective evolution algorithm (MOEA) is also developed to efficiently solve the multi-player non-cooperative game strategy in nonlinear stochastic jump diffusion system. The MOEA is employed to optimize the goals of all players and we propose the design procedure of a multi-objective (MO) optimal controller. After that, we can obtain non-cooperative strategies which can effectively achieve the best solution for each player at less sacrifice of the other goals. Finally, a simulation of a financial market is provided to illustrate the effectiveness of the proposed MO optimal control design procedure.

1 : Y. C. Ho, A. E. Bryson, Jr, and S. Baron, "Differential games and optimal pursuit-evasion strategies," IEEE Trans. Autom. Control, vol. 10, pp. 385--389, 1965.
2 : B. S. Chen, C. S. Tseng and H. J. Uang ,"Fuzzy differential game for nonlinear stochastic systems: suboptimal approach," IEEE Trans. Fuzzy Systems, Vol. 10, no. 22, pp. 222-233, 2002.
3 : H. Chen, R. Ye, X. Wang and R. Lu, "Cooperative control of power system load and frequency by using differential games," IEEE Trans. on Control System Technology, 23(3) 882-897, 2015.
4 : B. S. Chen and S. J. Ho, "The stochastic evolutionary game for a population of biological networks under natural selection," Evolutionary Bioinformatics, Vol. 10, no. 17, 00.17-38, 2014.
5 : J. Engwerda, "Necessary and sufficient conditions for Pareto optimal solutions of cooperative differential games," SIAM J. Control Optim., vol. 48, no. 6, pp. 3859--3881, 2010.
6 : T. Basar and G. J. Olsder, "Dynamic Noncooperative Game Theory." Academic Press, London, England, 1982.
7 : M. Maskery, et al, "Decentralized dynamic spectrum access for cognitive radios: cooperative design of a non-cooperative game," IEEE Trans. Commun., vol. 57, no. 2, pp. 459-469, 2009.
8 : Z. Han, Z. Ji, and K. J. R. Liu, "Non-cooperative resource competition game by virtual referee in multi-cell OFDMA networks," IEEE J. Sel. Areas Commun. vol. 25, no. 6, pp. 1079--1090, Aug. 2007.
9 : Z. He, H. Xu, and J. Wang, "Non-cooperative Differential Game based Load Balancing Algorithm in Radio-over-Fibre system," Cyberspace Technology (CCT 2014), International Conference on. Nov. 2014.
10 : B. S. Chen and C. H. Yeh, "Stochastic noncooperative and cooperative evolutionary game strategies of a population of biological network under natural selection," BioSystems, no. 162, pp. 90-118, 2017.
11 : R. Kamalapurkar, J. Klotz, and W. Dixon, "Concurrent learning-based online approximate feedback Nash equilibrium solution of N-player nonzero-sum differential games," IEEE/CAA J. Autom. Sin., vol. 1, no. 3, pp. 239--247, July 2014.
12 : T. Takagi and M. Sugeno, "Fuzzy identification of systems and its applications to modeling and control," IEEE Trans. Syst., Man, Cybern., vol. SMC-15, no. 1, pp. 116--132, Jan./Feb. 1985.
13 : C. S. Tseng, B. S. Chen and H. J. Uang, "Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model," IEEE Trans. Fuzzy Syst., vol. 9, no. 3,
14 : B. S. Chen, C. S. Tseng, H. J. Uang, "Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model," IEEE Trans. Fuzzy Syst., vol. 7, no. 5, pp. 571--585, Oct. 1999.
15 : C. S. Feng, B. S. Chen, and H. J. Uang, "Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model," IEEE. Trans. Fuzzy Syst. Vol 9, no. 3, pp.381-392, Jan, 2001.
16 : H. Zhang, J. Zhang, G.-H. Yang, and Y. Luo, "Leader-based optimal coordination control for the consensus problem of multiagent differential games via fuzzy adaptive dynamic programming," IEEE Trans. Fuzzy Syst., vol. 23, no. 1, pp. 152--163, Feb. 2015.
17 : B. K. Osendal and A. Sulem, "Applied Stochastic Control of Jump Diffusions," 2nd. Berlin, Germany: Springer, 2007.
18 : F. B. Hanson, "Applied Stochastic Processes and Control for Jump Diffusions: Modeling, Analysis, and Computation," Philadelphia, PA, USA: SIAM, 2007.
19 : D. Jacobson, "Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games," IEEE Transactions on Automatic Control, vol. 18, no. 2, pp. 124-131, 1973.
20 : B. S. Chen, W. Zhang, "Stochastic H₂/H_{∞} control with state-dependent noise," IEEE Trans. Autom. Control, vol. 49, no. 1, pp. 45-57, Jan. 2004.
21 : C. F. Wu, B. S. Chen and W. Zhang, "Multiobjective Investment Policy for a Nonlinear Stochastic Financial System: A Fuzzy Approach," IEEE Trans. Fuzzy Syst., vol. 25, no. 2, pp. 460--474, Apr. 2017.
22 : R. Cont and P. Tankov, "Financial Modeling with Jump processes," Boca Raton, FL, USA: CRC Press, 2004.
23 : K. Tanaka, and H. O. Wang, "Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach," John Wiley & Sons. Inc. 2001.
24 : S. Boyd, L. EI Ghaoui. E. Feron, and V. Balakrishnan, "Linear Matrix Inequalities in System and Control Theory," SIAM, Philadelphia 1994.
25 : P. Gahinet. A. Nemirovski. A. Laub, and M. Chilali. "The LMI Control Toolbox," Natick, MA: The Mathworks, Inc. 1995.
26 : K. Tanaka, and H. O. Wang, "Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach," John Wiley & Sons. Inc. 2001.
27 : R. L. Haupt and S. E. Haupt, "Practical Genetic Algorithms. 2nd ed." New York, NY, USA: Wiley, 2004.
28 : G. P. Liu, J. B. Yang, and J. F. Whidborne, "Multiobjective Optimizational Control," Hertfordshire, U.K.: RSP Ltd, 2002.
29 : L. Rachmawati and D. Srinivasan, "Multiobjective evolutionary algorithm with controllable focus on the knees of the Pareto front," IEEE Trans. Evol. Comput., Vol. 13, no.4, pp.810-824, Aug. 2009.
30 : K. Del, "Multi-objective Optimization Using Evolutionary Algorithms, 1st ed." Chichester, U.K.; Wiley, 2001.
31 : W.-Y. Chiu, "Multiobjective controller design by solving a multiobjective matrix inequality problem," IET Control Theory Appl., vol. 8, no. 16, pp. 1656--1665, Nov. 2014.
32 : A. Abraham, L. C. Jain, and R. Goldberg, "Evolutionary multiobjective optimization: Theoretical Advances and Applications," New York, NY, USA: Springer, 2005.
33 : K. Del, A. Pratap, S. Agarwal, and T. Meyarivan, "A fast and elitist multiobjective genetic algorithm: NSGA-II," IEEE Trans. Evol. Comput., Vol. 6, no. 2, pp.182-197, Apr. 2002.
34 : J. H. Ma and Y. S. Chen, "Study for the bifurcation topological structure and the global complicated character of a kind on nonlinear finance system," App. Math. Mech., Vol. 22, no. 11, pp.1240-1251, 2001.
35 : A. Isidori and A. Astolfi, "Disturbance attenuation and H_{∞} control via measurement feedback in nonlinear svstems", IEEE Transactions on Automatic Control, vol. 37, no. 9, pp. 1283-1293, 1992.
36 : W. Zhang, L. Xie and B. S. Chen, "Stochastic H₂/H_{∞} control: A Nash Game Approach," CRC Press, Boca Taton London England, 2017.
37 : B. S. Chen and W. Zhang, "Stochastic H₂/H_{∞} control with state dependent noise," IEEE Trans. Automatic Control, Vol.49, no.1, pp.45-57, Jan.2004.