Modeling,Analysis And Control Of Networked Evolutionary Games With Time Delays

Based on the algebraic state space method,this paper investigates the modeling,analysis and control of networked evolutionary games(NEGs)with time delays.With the help of the semi-tensor product of matrices,several delayed NEGs are transformed into algebraic formulations,including uniform time delays,non-uniform time delays,state-dependent time delays,probabilistic time delays,and switched time delays.Based on the algebraic equations,the convergence,stability,controllability and optimal control of NEGs with time delays are studied.Specifically:Firstly,the deterministic algebraic formulations are established for NEGs with deterministic time delays,i.e.,uniform time delays,non-uniform time delays,and state-dependent time delays.For NEGs with probabilistic time delays,the probability transition matrix of augmented profiles is derived,and the algebraic formulation following probability is established.For NEGs with switched time delays,the subsystems with deterministic switching signals are calculated,and the algebraic formulation switching among several deterministic subsystems is established.Secondly,the convergence and stability of NEGs with several kinds of time delays are investigated.Based on the algebraic formulation of NEGs with deterministic time delays,the transition matrix of augmented profiles is calculated,and the necessary and sufficient conditions for the local stability and global stability of the system are obtained.Based on the algebraic formulation of NEGs with probabilistic time delays,the transition probability of profiles is calculated,and the necessary and sufficient conditions of set stability are deduced.Based on the algebraic formulation of NEGs with switched time delays,the criterion of profile reachability with switching signals is proposed,and the necessary and sufficient conditions of local convergence and global convergence are deduced.In order to describe the dynamic stability of NEGs with time delays,the concept of the evolutionarily stable profile is proposed,and it is proved to be a special local stable point.Based on the reachability of profile and local stability,the existing condition of the evolutionarily stable profile is deduced.Thirdly,the controllability and set stabilization of NEGs with time delays are studied.Driving the game evolution by controlling part of the players,the state profile transition matrix and the control profile feedback matrix are established.Based on these,the criterion of controllability and the necessary and sufficient conditions of set stabilization are deduced,respectively.In order to design the set stabilization controller,the truth matrix method is proposed.By solving logical equations,the control profile feedback matrix is designed,under which the NEGs with time delays are stabilized at the target set.Finally,the infinite horizon optimal control problem is presented for NEGs with time delays.Using the matrix decomposition method and limit theory,the limiting payoff vector is calculated for infinitely repeated NEGs and corresponding algebraic properties are derived.In order to design a controller to realize the infinite horizon optimal payoff,the concept of path payoff is proposed.By calculating the path payoff matrix,the relationship between the limiting payoff vector and the path payoff matrix is derived,and an algorithm is designed to find the optimal controller.