On stability of time-varying multidimensional linear systems

A large class of dynamical systems can be modelled by differential equations of the form (X) double over dot + D(X) over dot + K(t)X = 0, where X is an element of R-n, D, K(t) is an element of M-n(R). The stability problem of this form has been investigated by a number of researchers. For the general results and concepts of stability of the system see Bellman (1953), Lasalle (1968) and Coppel (1965). For the applications in the circuits with variable capacity, the vibrations of structures, and robotics control, see Richards (1983), Huseyin (1978) and Arimoto and Miyazaki (1986), respectively. In Sanderberg (1965), Sanderberg used frequency domain approach to achieve some criteria for the stability of the system. Recently, Shrivastava and Pradeep (1985) applied Lyapunov's theory to the system to derive several theorems. Hsu and Wu (1991) applied the same theory to estimate the margin of asymptotical stability. A linear time-varying system may have complicated behavior in the solutions can be seen in Galbrath et al. (1965). If K is changing periodically, then without the damping D, the system may have parametric resonance and some solutions may grow up to infinity exponentially. So, in order to stabilize the system, certain amount of damping is needed. In this note, we analyze the damping matrix D and estimate the minimum amount of the damping to make the system exponentially stable.