A case-study of strict-Lyapunov-function design for adaptive-control systems with non-smooth regressors

We revisit the problem of designing a strict Lyapunov function for systems appearing in model-reference adaptive control. Even though there exist well-established conditions for uniform exponential stability (for the linear case) and uniform global asymptotic stability (for the nonlinear case) in the literature, and some works provide strict Lyapunov functions, most of them are based on the assumption that the regressor is smooth and its derivative admits a (known) uniform bound. Our main result consists in relaxing this assumption, which is notably useful in tracking control tasks, where it is required to track arbitrarily rapid reference trajectories. Our main statement is formulated for linear time-varying systems with regressors of dimension one, but we show how it applies in the analysis of nonlinear systems; notably in adaptive control and adaptive observer design.