On stability of time-varying multidimensional linear systems

A large class of dynamical systems can be modelled by differential equation s 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 stabilit y problem of this form has been investigated by a number of researchers. Fo r the general results and concepts of stability of the system see Bellman ( 1953), Lasalle (1968) and Coppel (1965). For the applications in the circui ts with variable capacity, the vibrations of structures, and robotics contr ol, see Richards (1983), Huseyin (1978) and Arimoto and Miyazaki (1986), re spectively. In Sanderberg (1965), Sanderberg used frequency domain approach to achieve some criteria for the stability of the system. Recently, Shriva stava 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 co mplicated 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 h ave parametric resonance and some solutions may grow up to infinity exponen tially. 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 mini mum amount of the damping to make the system exponentially stable.