Vector-valued quadratic forms in control theory

Francesco Bullo, Jorge Cortés, Andrew D. Lewis and Sonia Martínez
Workshop on Open Problems in Mathematical Systems and Control Theory, Symposium on Mathematical Theory of Networks and Systems, August 12-16, 2002, Notre Dame, Indiana, USA

Abstract:

Given two real vector spaces U and V, and a symmetric bilinear map B: U x U -> V, let Q_B be its associated quadratic map. The problems we consider are as follows: (i) are there necessary and sufficient conditions, checkable in polynomial-time, for determining when Q_B is surjective?; (ii) if Q_B is surjective, given v belonging to V is there a polynomial-time algorithm for finding a point u in the inverse image of v by Q_B?; (iii) are there necessary and sufficient conditions, checkable in polynomial-time, for determining when B is indefinite? We present an alternative formulation of the problem of determining the image of a vector-valued quadratic form in terms of the unprojectivised Veronese surface. The relation of these questions with several interesting problems in Control Theory is illustrated.

File: main.pdf

Bib-tex entry:

@InProceedings{FB-JC-ADL-SM:02,
author = {F. Bullo and J. Cort\'es and A.D. Lewis and S. Mart{\'\i}nez},
booktitle = {Workshop on Open Problems in Mathematical Systems and Control Theory},
title = {Vector-valued quadratic forms in control theory},
year = {2002},
address = {South Bend, Indiana},
month = {August}
}