A Lie-algebraic analysis of the problem of absolute stability

The absolute stability problem (ASP) entails determining a critical parameter value for which a feedback system, composed of an nth-order linear system and a sector-bounded nonlinear function, loses its stability. The ASP is one of the oldest open problems in the theory of stability and control. Recently, it is attracting considerable interest, as solving it is equivalent to providing a necessary and sufficient condition for the stability of linear switched systems under arbitrary switching. Pyatnitsky pioneered the most promising approach for addressing the ASP. His approach is based on using optimal control techniques for characterizing the “most destabilizing” sector-bounded nonlinear function. This is equivalent to characterizing the “most destabilizing” switching law for a linear switched system under arbitrary switching. In this paper, we develop a new approach to the ASP which is based on a Lie-algebraic analysis of the switching function that determines the optimal control. We show that the finiteness of the associated Lie algebra implies that the switching function itself is the solution of a switched linear system of order at most n². Furthermore, the switching function has a special and symmetric structure. This makes it possible to obtain an explicit analytic expression for the switching function for low orders of n. We demonstrate this using two examples.