Noncooperative and Cooperative Multiplayer Minmax H∞ Mean-Field Target Tracking Game Strategy of Nonlinear Mean Field Stochastic Systems With Application to Cyber-Financial Systems

In this study, we investigate multi-player noncooperative minmax $H_{\infty }$ target tracking game strategy with conflicting target strategies and cooperative $H_{\infty }$ target tracking game strategy with common target strategy of nonlinear mean-field stochastic jump diffusion (MFSJD) system with external disturbance. Due to the nonlinear terms and mean-field (average) behaviors in the stochastic nonlinear MFSJD system and minmax $H_{\infty }$ payoff function, the multi-player noncooperative and cooperative minmax $H_{\infty }$ mean-field game strategy of nonlinear MFSJD system are more difficult than the linear MFSJD system and conventional nonlinear stochastic system. To avoid solving complex Hamilton Jacobi Isaacs inequalities (HJIIs) of multi-player noncooperative minmax $H_{\infty }$ mean-field target tracking game strategy of nonlinear MFSJD system, the nonlinear MFSJD system is interpolated by a set of local linearized MFSJD system through smoothing functions by the global linearization method. Then the multi-player noncooperative minmax $H_{\infty }$ nonlinear mean-field target tracking game strategy design can be transformed to a linear matrix inequalities (LMIs)-constrained multi-objective optimization problem (MOP). The LMIs-constrained MOP could be efficiently solved by the help of the proposed LMIs-constrained multi-objective evolution algorithm (MOEA). We can prove that the Pareto optimal solution of LMIs-constrained MOP is the Nash equilibrium solution of noncooperative minmax $H_{\infty }$ mean-field target tracking strategy of nonlinear mean-field MFSJD system. Further, the cooperative minmax $H_{\infty }$ mean-field common target tracking strategy of nonlinear mean-field stochastic system is reduced to an LMIs-constrained single-objective optimization problem (SOP). Finally, two simulation examples of cyber-financial mean-field systems are given to illustrate the design procedure and compare the efficacies of the proposed noncooperative and cooperative minmax $H_{\infty }$ mean-field target tracking strategy of nonlinear MFSJD systems.