一类线性时变群体系统稳定性的代数判据

Algebraic Criterion for the Stability of a Class of Linear Time-Varying Swarm Systems

针对在判断群体系统的稳定性时没有一般的方法和程序构造Lyapunov函数这个难点,利用矩阵范数,孤立子系统的矩阵指数函数与比较原理提出了一类线性时变群体系统平凡解一致稳定,所有解一致有界的充分条件。方便此类群体系统的稳定性分析,为研究其他群体系统稳定性的代数判据提供了理论基础。可以在此基础上,进一步研究一类非线性群体系统稳定性的代数判据。同时,还给出了具体算例,说明所提方法的正确性。此代数判据应用简便,灵活,适于实际应用。

In view of the difficulty that there is not the general method and the procedure to structure the Lyapunov function, when judge the stability of the swarm systems , by the matrix norm, the matrix exponential function of the subsystem and the comparison principle, a sufficient condition of the uniform stability of the trivial solution and the uniform boundedness of all solutions for a class of linear time-varying swarm systems is proposed. The method makes the analysis of the stability for this class of swarm system more con- venient. The rationale for studying the algebraic criterion of the other swarm systems is provided in this method. The algebra criterion for a class of non-linear swarm systems can be further studied. Simultaneously, the concrete example has been given to illustrate the accuracy of the proposed method. The algebraic criterion which is simple and flexible is suitable for the practical application.