Analysis And Control Of Finite Evolutionary Games With Applications Via The Semi-tensor Product Method

As an important part of game theory,evolutionary game theory has received considerable attention,because of its wide background in biological systems,economical systems,social systems,and engineering science,etc.From dynamic evolution perspective,finite evolutionary games can be equivalent to finite valued logical dynamic systems.As a powerful tool to study logical dynamic systems,semi-tensor product(STP)of matrices can also be used to study finite evolutionary games.Based on this method,this paper firstly studies the stability and stabilization of evolutionary games with time delays.Then the intermittent control for demand-side management of a class of networked smart grid is considered.Finally,the dynamics and control of evolutionary congestion games are investigated.The main contents are as follows.1.The stability and stabilization of evolutionary games with time delays are studied.First,the algebraic expression of evolutionary games with time delays is given by the STP method.Then a necessary and sufficient condition for the global stability of delayed evolutionary games is obtained by using Lyapunov function.We also show that delayed evolutionary potential games can converge to a pure Nash equilibrium under certain strategy updating rule(SUR).Finally,the global stabilization problem of controlled evolutionary games with time delays is investigated and a necessary and sufficient condition is provided to assure the stabilization.2.The demand-side management problem and intermittent control for a class of networked smart grid are investigated.First,the networked smart grid is modelled into a special networked evolutionary game.Then,to minimize the overall cost of all the communities and reduce the control execution time,intermittent control is designed for some communities,in which these players work as controllers at some special strategy profiles.By designing the intermittent open-loop control and intermittent state feedback control,the smart grid can converge to the desired strategy profile globally.Algorithms that can compute the state feedback controllers are given.Finally,a numerical example is worked out to illustrate the effectiveness of the developed theoretical results.3.The evolutionary dynamics and intermittent control of a class of congestion games are considered.First,the congestion game is expressed into the algebraic form by STP.Then,with the parallel myopic best response adjustment(MBRA),the rigorous dynamic equation of evolutionary congestion games is established and some properties are obtained.Then to make the congestion game dynamics converge to Nash equilibriums rather than the limit cycle whose length is greater than one,some players act as controllers and intermittent control is designed for them.By designing the intermittent open-loop control and state feedback control,respectively,two necessary and sufficient conditions are provided to assure the stabilization of the congestion game.A numerical example is worked out to illustrate the effectiveness of the theoretical results.