Estimating Domains Of Attraction Based On Moment And Sum-of-squares Optimization

Estimating the domain of attraction (DOA) of a dynamical system is an important subject of the theory of stability. In many engineering fileds, for example, electric systems and chemical reactors, for many non-linear dynamical systems, for the work in security, we must know the DOA of the systems. This paper studies estimating the DOA of non-linear autonomous systems using the related mathematical theory, we suggest two algorithms, the LMI method based on moment theory and sum-of-squares (SOS) optimization algorithm.Infectious disease is a very common phenomenon. We establish the infection mathematical model using the dynamics methods, and qualitative analysis and quantitative analysis through mathematical model have been obtained some results. Knowing deeply about DOA of this system can direct us to determine and forecast the development trend of infection. This paper studies a class SIR infection model, analyses its stability, at the same time, estimates its DOA using the primal and the dual optimization algorithm of moment theory and SOS optimization algorithm.SOS optimization algorithm offers a dynamical iterative algorithm, it can work out an optimization Lyapunov function, this dynamical interact manner is more advantage than LMI method, because LMI must know the Lyapunov function beforehand. The simulation shows SOS optimization algorithm can obtain a larger DOA. SOS optimization algorithm can deal with the product of unknown quantities, i.e. non-linear problems, so it will be an efficacious tool to solve the control problems following the LMI method.