Stability of Linear Systems With Parametric Excitation

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Abstract
A Lyapunov-type approach is used to develop sufficient asymptotic stability conditions for linear systems with time-varying coefficients. In particular, it is shown that parametric disturbances of high frequency cannot create instability in an already asymptotically stable system. Also it is shown that slowly varying parametric disturbances will not cause instability if the system matrix is a stability matrix for all values of time. The results are applied to the Mathieu equation to illustrate the character of the theorems. This example is chosen because of the availability of its exact stability boundaries.

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