Finding non-polynomial positive invariants and lyapunov functions for polynomial systems through Darboux polynomials

In this paper, we focus on finding positive invariants and Lyapunov functions to establish reachability and stability properties, respectively, of polynomial ordinary differential equations (ODEs). In general, the search for such functions is a hard problem. As a result, numerous techniques have been developed to search for polynomial differential variants that yield positive invariants and polynomial Lyapunov functions that prove stability, for systems defined by polynomial differential equations. However, the systematic search for non-polynomial functions is considered a much harder problem, and has received much less attention. In this paper, we combine ideas from computer algebra with the Sum-Of-Squares (SOS) relaxation for polynomial positive semi-definiteness to find non polynomial differential variants and Lyapunov functions for polynomial ODEs. Using the well-known concept of Darboux polynomials, we show how Darboux polynomials can, in many instances, naturally lead to specific forms of Lyapunov functions that involve rational function, logarithmic and exponential terms.We demonstrate the value of our approach by deriving non-polynomial Lyapunov functions for numerical examples drawn from the literature.