时效网络最小控制能量的研究

Research on the Minimum Control Energy of Temporal Networks

研究了系统在实现能控时,时效网络的最小控制能量问题,得出最小控制能量的上下界仅由其相关的Gram矩阵的最小和最大特征值决定。由此给出了计算最小控制能量下界的方法,通过分析得出时效网络最小控制能量下界与驱动节点个数和瞬时状态个数无关。当切换时间h趋于0时,时效网络最小控制能量的下界趋向于1/2h,即E^(∗)~(1/2h),与静态网络相比,得出时效网络实现能控时的最小控制能量总小于静态网络,并给出了其之间的数值关系。当系统实现控制目的的终止时间tf趋向于0时,静态网络最小控制能量的下界趋向于1/tf。当tf=h时,可以得出,时效网络最小控制能量的下界是静态网络的1/2。

In this paper,the minimum control energy problem of temporal networks is studied when the system is controlled.The upper and lower bounds of the minimum control energy are determined only by the minimum and maximum eigenvalues of the associated Gram matrix.The method of calculating the lower bound of the minimum control energy is given.Through analysis,it is concluded that the minimum control energy of temporal networks are independent of the number of driving nodes and the number of snapshots.When the switching time h approaches 0,the minimum control energy of temporal networks tend to 1/2h,namely E∗~(1/2h).Compared with static networks,the minimum control energy of temporal networks for achieving controllability are always smaller than that of static networks,and the numerical relationship between them is given.When tf which is the termination time of the system for control purposestends to 0,the lower bound of the minimum control energy of static networks tends to 1/tf.When tf=h,it can be concluded that the lower bound of the minimum control energy of temporal networks is 1/2 of that of static networks.