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In this work the stability of perturbed linear time-varying systems isstudied. The main features of the problem are threefold. Firstly, thetime-varying dynamics is not required to be continuous but allowed to havejumps. Also the system matrix is not assumed to be always Hurwitz. In addition,there is nonlinear time-varying perturbation which may be persistent. We firstpropose several mild regularity assumptions, under which the total variationsof the system matrix and its abscissa are well-defined over arbitrary timeinterval. We then state our main result of the work, which requires thecombined assessment of the total variation of the system matrix, the measurewhen the system is not sufficiently "stable" and the estimate of theperturbation to be upper bounded by a function affine in time. When thiscondition is met, we prove that the neighborhood of the origin, whose sizedepends on the magnitude of the perturbation, is uniformly globallyexponentially stable for the system. We make several remarks, connecting ourresults with the known stability theory from continuous linear time-varyingsystems and switched systems. Finally, a numerical example is included tofurther illustrate the application of the main result.

Unified stability criteria for perturbed LTV systems with unstable instantaneous dynamics