变系数动力系统的运动稳定性

<正> §1 序言 随时间t变化的变系数动力系统的运动稳定性在工程和物理中是经常遇到的叫题,例如依靠尾翼来稳定的火箭弹在火箭推力起作用的过程中的运动状态;飞机的稳定与控制.

ON THE STABILITY OF MOTION OF DYNAMICAL SYSTEMS WITH TIME-VARYING COEFFICIENTS

For a linear system of differential equations with constant coefficients. dx/dt=Ax(1)It. is well-known thai, if all eigenvalues of A have negative real parts, then the trival solution of (1) is asymptotieally stable.However, the problem is much more difficult for a linear time-varying system dx/dt=A(t)x in the relation between the signs of Re(λ_i(t)) of the eigenvalues of A(t) and the stability of the dynamical systems. Examples of all kinds of the combinations are given in this paper to show its eomplexity. But if the rate of change of the elements a_(ij)(t) of A(t) is sufficiently slow, by using the method of Lyapunov funetion, the problem ean be solved thoroghly. In addition, an explieit bound for the a_(ij)'(t) is given, no matter what form matrix A(t) has.