A Numerical Algorithm for Solving the Absolute Stability Problem in R3

The problem of absolute stability is one of the oldest open problems in the theory of control. For low-order systems, the most general results were obtained by Pyatnitskiy and Rapoport. They derived an implicit characterization of the "most destabilizing" nonlinearity using the maximum principle. In this paper, we show that their approach yields a simple and efficient numerical scheme for solving the problem in the case of third-order systems. This allows the determination of the critical value where stability is lost in a tractable and accurate fashion. This value is important in many practical applications and we believe that it can also be used to develop a deeper theoretical understanding of this interesting problem.