### Jorge Cortés

#### Professor

Vector-valued quadratic forms in control theory

F. Bullo, J. Cortés, A.D. Lewis, S. Martinez

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.

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Mechanical and Aerospace Engineering,
University of California, San Diego

9500 Gilman Dr,
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Ph: 1-858-822-7930

Fax: 1-858-822-3107

cortes at ucsd.edu

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jorgilliyo