  In this work the stability of perturbed linear time-varying systems is
studied. The main features of the problem are threefold. Firstly, the
time-varying dynamics is not required to be continuous but allowed to have
jumps. Also the system matrix is not assumed to be always Hurwitz. In addition,
there is nonlinear time-varying perturbation which may be persistent. We first
propose several mild regularity assumptions, under which the total variations
of the system matrix and its abscissa are well-defined over arbitrary time
interval. We then state our main result of the work, which requires the
combined assessment of the total variation of the system matrix, the measure
when the system is not sufficiently "stable" and the estimate of the
perturbation to be upper bounded by a function affine in time. When this
condition is met, we prove that the neighborhood of the origin, whose size
depends on the magnitude of the perturbation, is uniformly globally
exponentially stable for the system. We make several remarks, connecting our
results with the known stability theory from continuous linear time-varying
systems and switched systems. Finally, a numerical example is included to
further illustrate the application of the main result.
