# American Institute of Mathematical Sciences

October  2015, 20(8): 2383-2417. doi: 10.3934/dcdsb.2015.20.2383

## Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares

 1 School of Matter, Transport and Energy, Arizona State University, 650 E. Tyler Mall - GWC 531, Tempe, 85281, United States 2 School of Matter, Transport and Energy, Arizona State University, 501 Tyler Mall - ECG 301, Tempe, 85281, United States

Received  June 2014 Revised  December 2014 Published  August 2015

In this paper, we explore the merits of various algorithms for solving polynomial optimization and optimization of polynomials, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation Linear Techniques, Blossoming and Groebner basis methods, our main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and Handelman's theorem. We first formulate polynomial optimization problems as verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's algorithm, Bernstein's algorithm and Handelman's algorithm reduce the intractable problem of feasibility of semi-algebraic sets to linear and/or semi-definite programming. We apply these algorithms to different problems in robust stability analysis and stability of nonlinear dynamical systems. As one contribution of this paper, we apply Polya's algorithm to the problem of $H_\infty$ control of systems with parametric uncertainty. Numerical examples are provided to compare the accuracy of these algorithms with other polynomial optimization algorithms in the literature.
Citation: Reza Kamyar, Matthew M. Peet. Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2383-2417. doi: 10.3934/dcdsb.2015.20.2383
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