Stability of a dynamical system with respect to finite perturbations can be established by finding a Lyapunov function, or a Lyapunov functional in case of infinite dimensional systems. However, there is no general systematic method for constructing Lyapunov functions - the discovery of such a function is dependent on the ingenuity and creativity of the investigator.
Fortunately, a recent breakthrough in control theory (Parrilo, 2000) has provided a constructive method for generating Lyapunov functions for systems whose dynamics can be described by polynomial functions. This method is based on sum-of-squares (SOS) optimization, which reduces the problem of finding a Lyapunov function for a polynomial system to one of constructing a polynomial function that satisfies a selection of algebraic conditions. Using a number of key results from semialgebraic geometry, in particular the Positivestellensatz theorem, the resulting problem can be reformulated as an optimization problem in the form of a Semi-Definite Programme (SDP). This is particularly promising since SDP optimization problems are convex and tractable (e.g. solvable in a number of operations that is a polynomial function of the problem size), and theoretical and algorithmic research into solving such problems is extremely active. A variety of well-supported software codes for solving such problems are freely available, both for SOS problems in particular and SDP problems in general.
More recently, research has focussed on the efficient solution of semidefinite programming problems arising from SOS problems, with particular emphasis on robust optimization and exploitation of structure and sparsity in large problems. As a result of these advances, the SOS approach has found extensive applications in stability analysis, control theory and many other fields, including applications in aeronautics. The use of semidefinite programming also has a long history in systems analysis and control, since many control problems can be formulated and solved this way.
In parallel with these developments, recent work by Rantzer (2001) has provided an alternative means of demonstrating stability of a dynamic system, using a so-called dual Lyapunov or density function. Crucially, using this method it is also possible to compute simultaneously both a controller for a nonlinear system and a dual Lyapunov function certifying its stability, which is generally not possible using Lyapunov methods. For problems with polynomial dynamics, this can be achieved by solving a single convex optimization problem (again an SDP) using the SOS framework.