A decomposition of multivariable forms and its applications

Pablo Parrilo, Caltech, Control & Dynamical Systems

In the last few years, linear matrix inequalities (LMI) based methods have demostrated an amazing versatility in the systems and control area. The numerous applications of semidefinite programming to basic applied mathematics problems (mainly, powerful relaxations of NP-hard combinatorial optimization problems) show that this trend is bound to continue in the next few years.

In this talk, we will show that some of the convenient characteristics of LMI-based methods can be extended to a class of multivariable forms. The main tool is the use a computationally tractable sufficient condition for positivity of a function, namely the existence of a ``sum of squares'' representation. By using an extended set of variables and redundant constraints, it is shown that the conditions can be written as linear matrix inequalities in the unknown parameters.

We will present the application of the results to some concrete problems, including Lyapunov stability of nonlinear systems described by polynomial vector fields, stronger sufficient conditions for copositivity of a matrix, and improved tests for robustness analysis problems.