Title: Explicit Commutativity Conditions for Second-order Linear Time-Varying Systems with Non-Zero Initial Conditions,
Abstract: Although the explicit commutativitiy conditions for second-order linear
time-varying systems have been appeared in some literature, these are all for
initially relaxed systems. This paper presents explicit necessary and
sufficient commutativity conditions for commutativity of second-order linear
time-varying systems with non-zero initial conditions. It has appeared
interesting that the second requirement for the commutativity of non-relaxed
systems plays an important role on the commutativity conditions when non-zero
initial conditions exist. Another highlight is that the commutativity of
switched systems is considered and spoiling of commutativity at the switching
instants is illustrated for the first time. The simulation results support the
theory developed in the paper.