Global Dynamics And Steady States Of Mean Field Games With Quadratic Hamiltonians

Game is the process in which rational decision-makers under constraints choose and carry out their strategies to minimize cost or maximize utility.Since this process is vital for the analysis of economic activities and military affairs,such as the price formation in economic market,the armament race between the United States and the Soviet Union,etc,the theoretical subject,Game Theory,is established to understand it.However,involving with a huge number of participants,it is complicated to utilize the classical game theory to model.In 2007,Jean Michel Lasry and Pierre Louis Lions proposed a class of parabolic PDE systems which have relatively simple structures,namely mean field models,to describe the evolution of single “player” density and the other one in the “mean-field” sense when there exist a large number of homogeneous decision-makers in the environment.This thesis is devoted to investigate the global dynamics and steady states of mean field models with quadratic Hamiltonians.When the growth of cost is in compliance with the power law,we first prove the local existence of the classical solution in system,then show its global existence and uniform boundedness via the construction of Lyapunov functionals.Meanwhile,we use some inequalities to derive that the constant steady state is a global attractor.When the cost satisfies the decay property,we employ the bifurcation theory to show the existence of nonconstant positive steady states,meanwhile study the asymptotic behavior of monotone one,then utilize the singular perturbation method to construct its asymptotic expansion and show the uniqueness.Finally,numerical simulation is shown in this thesis to illustrate and verify the theoretical results.