R-composition of Lyapunov functions

This paper introduces the use of R-functions to compose (R-composition) basic simple Lyapunov functions, like the conventional quadratic ones, to obtain a larger variety of functions. R-functions represent the natural extension of Boolean operators to real-valued functions and provide the basic tools to compute the analytic expression of intersection and union operations in a geometric setting. In the framework of Lyapunov approaches to prove stability of a dynamical system, the union of Lyapunov functions computed through the R-function approach is still a Lyapunov function. Moreover, as each Lyapunov function defines the shape and the orientation of a correspondent geometric Largest Estimate of the Domain of Attraction (LEDA), then the LEDA associated to the union of several Lyapunov functions corresponds to the union of the single LEDAs. R-composition of Lyapunov functions thus corresponds to a non-conventional Lyapunov function which can be used to improve the estimate of the Region of Asymptotic Stability (RAS) and at the same time to introduce more freedom in the choice of the shape of the correspondent level sets, that in general are non-convex. An example of the R-composition of Lyapunov functions is illustrated to solve a classic RAS estimation problem.