Determination of the optimal quadratic Lyapunov function for nonlinear autonomous systems via interval arithmetic

The estimation of the Domain of Attraction (DA) is an important topic in investigation of system stability. Using Lyapunov´s conditions leads to a dependency of the chosen Lyapunov function and the size of the estimation. This paper presents a method, which delivers the coefficients of the optimal quadratic Lyapunov function (OQLF) for a given system. Additionally the method determines the largest estimation of the DA. Therefore, the coefficients of the quadratic Lyapunov function are intervals. The coefficients are optimized by an iteratively outer optimization algorithm based on interval arithmetic. An inner optimization algorithm delivers an estimation for the DA´s size of all QLF which can be constructed from the coefficient intervals. This yields the largest estimation of the DA as well as OQLF for the given system.