  We propose a framework to analyze stability for a class of linear
non-autonomous hybrid systems, where the continuous evolution of solutions is
governed by an ordinary differential equation and the instantaneous changes are
governed by a difference equation. Furthermore, the jumps are triggered by the
influence of an external hybrid signal. The proposed framework builds upon a
generalization of the well-known persistency of excitation (PE) and uniform
observability (UO) notions to the realm of hybrid systems. That is, we
establish conditions, under which, hybrid PE implies hybrid UO and, in turn,
uniform exponential stability (UES) and input-to-state stability (ISS). Our
proofs rely on an original statement for hybrid systems, expressed in terms of
Lp bounds on the solutions. We demonstrate the utility of our results on
generic adaptive estimation problems. The first one concerns the so-called
gradient systems, reminiscent of the popular gradient-descent algorithm. The
second one pertains to designing adaptive observers/identifiers for a class of
hybrid systems that are nonlinear in the input and the output, and linear in
the unknown parameters. In both cases, we illustrate through examples that the
proposed hybrid framework succeeds when the classic purely continuous- or
discrete-time counterparts fail.
