The Controllability And Optimal Control Of Incomplete Boolean Networks With Their Applications In Game Theory

The analysis and control of the Boolean network is a hot research topic in control theory.It is not only a powerful tool to describe the gene regulatory network,but also can be used to describe the dynamics of a double strategy repeated game with finite players.In the actual network,some control-states can be disadvantageous or even dangerous(such as the deterioration of diseases or the metastasis of a cancer in the gene regulatory network,and the bankruptcy of players in the game,etc.).In order to avoid these unfavorable situations,it is necessary to establish an incomplete Boolean network with control-state constraints,which can provide better theoretical support to solve such problems.By resorting to the semi-tensor product technique,this paper investigates the controllability and optimal control of several kinds of incomplete Boolean control networks.Moreover,the obtained results are used to study the strategy optimization for a class of networked evolutionary games with bankruptcy mechanism.The main contents of this paper are listed as follows:1.The controllability and optimal control of high order incomplete Boolean control networks with impulse effects are analyzed.By constructing a state transition matrix,the order incomplete Boolean control network is expressed as an algebraic form.Based on which,several necessary and sufficient conditions for the controllability of the incomplete logical system are presented.Moreover,the Mayer-type optimal control issue is studied and the optimal control design algorithms are established.2.The problems of robust-output-controllability and robust optimal output control for incomplete Boolean control networks with disturbance inputs are studied.First,the model of disturbed incomplete Boolean control network is established.Then,by giving the algebraic form of this incomplete logical system via semi-tensor product,several necessary and sufficient conditions for the robust output controllability are obtained.Finally,the Mayer-type robust optimal output control issue is discussed and an algorithm is established to find a control scheme which can minimize the cost functional regardless of the effect of disturbance inputs.3.The optimal control problem of switched singular Boolean control networks with state and input constraints is investigated.First,the general switched singular Boolean control networks are introduced,and with semi-tensor product method,the algebraic form of a switched singular Boolean control network is given.Based on the algebraic form,a necessary and sufficient condition is given for the uniqueness of solution of the system under any switching signal and any control strategy.Then,a switched singular Boolean control network with state and input constraints is converted into an equivalent unconstrained switched Boolean control network.Finally,under certain assumption and using an analogous needle variation,a necessary condition for the existence of optimal control is given.4.The strategy optimization for a class of networked evolutionary games with bankruptcy mechanism is investigated.First,using the algebraic form,the strategy optimization problem of the given networked evolutionary games is transformed into the control problem of incomplete logical network with control-state avoiding set.Then,a necessary and sufficient condition is proposed to check whether an initial state can reach to the optimal state by proper control sequences,while avoiding the forbidden controlstates.Moreover,a constructive procedure is presented for the design of state feedback controller to maximize the long-run average payoff of the the given networked evolutionary games while avoiding any player going bankrupt.