Lyapunov functions for time varying systems satisfying generalized conditions of Matrosov theorem

The classical Matrosov theorem concludes uniform asymptotic stability of time varying systems via a weak Lyapunov function (positive definite, decrescent, with negative semi-definite derivative along solutions) and another auxiliary function with derivative that is strictly non-zero where the derivative of the Lyapunov function is zero [10]. Recently, several generalizations of the classical Matrosov theorem that use a finite number of Lyapunov-like functions have been reported in [5]. None of these results provides a construction of a strong Lyapunov function (positive definite, decrescent, with negative definite derivative along solutions) that is a very useful analysis and controller design tool for nonlinear systems. Inspired by generalized Matrosov conditions in [5], we provide a construction of a strong Lyapunov function via an appropriate weak Lyapunov function and a set of Lyapunov-like functions whose derivatives along solutions of the system satisfy a particular triangular structure. Our results will be very useful in a range of situations where strong Lyapunov functions are needed, such as robustness analysis and Lyapunov function based controller redesign.