Strict Lyapunov functions for time-varying systems with persistency of excitation

We study the stability of the origin for a class of linear time-varying systems with a drift that may be divided in two parts. Under the action of the first, a function of the trajectories is guaranteed to converge to zero; under the action of the second, the solutions are restricted to a periodic orbit. Hence, by assumption, the system's trajectories are bounded. Our main results focus on two generic case studies that are motivated by common nonlinear control problems: model-reference adaptive control, control of nonholonomic systems, tracking control problems, to name a few. Then, based on the standing assumption that the system's dynamics is persistently excited, we construct a time-dependent Lyapunov function that has a negative definite derivative. Our main statements may be regarded as off-the-shelf tools of analysis for linear and nonlinear time-varying systems.