Based on the seminal work of Naum Zuselevich Shor (Institute of Cybernetics,
Kiev) in the 1980s [1, 2, 3], the theory of positive polynomials lends new
theoretical insights intoawide range of control and optimization problems.
Positive polynomials can be used to formulate a large number of problems
in robust control, non-linear control and non-convex optimization. Only very
recently it has been realized that polynomial positivity conditions can be formulated
efficiently in terms of Linear Matrix Inequality (LMI) and Semidefinite
Programming (SDP) problems. In turn, it is now recognized that LMI
and SDP techniques play a fundamental role in convex optimization, see e.g.
the plenary talk by Stephen Boyd at the 2002 IEEE Conference on Decision
and Control or the successful Workshop on SDP and robust optimization organized
in March 2003 by the Institute of Mathematics and its Applications
at the University of Minnesota in Minneapolis. For the above reasons, the
joint use of positive polynomials and LMI optimization provides an extremely
promising approach to difficult control problems.