On the characterization of the solution set of polynomial systems via LMI techniques

The problem addressed in this paper is the computation of the solution set for systems of polynomial equations, a key issue in several system analysis and control problems. A new approach is presented, which represents a possible alternative to well-known techniques, based on algebraic geometry and homotopy methods. The basic idea is to characterize the solution set in terms of the kernel of a symmetric matrix, associated to a suitable quadratic homogeneous form. This matrix is obtained via a Linear Matrix Inequality (LMI) optimization problem. The actual computation of the solution set can be performed quite easily, provided that the dimension of the kernel does not exceed a prescribed value. It is shown that this value turns out to be quite large, so that the proposed procedure can be applied to a fairly wide variety of polynomial systems.