# The Convexity of Quadratic Maps and the Controllability of Coupled Systems

 Title: The Convexity of Quadratic Maps and the Controllability of Coupled Systems Author: Sheriff, Jamin Lebbe Citation: Sheriff, Jamin Lebbe. 2013. The Convexity of Quadratic Maps and the Controllability of Coupled Systems. Doctoral dissertation, Harvard University. Full Text & Related Files: Sheriff_gsas.harvard_0084L_11019.pdf (870.3Kb; PDF) Abstract: A quadratic form on $$\mathbb{R}^n$$ is a map of the form $$x \mapsto x^T M x$$, where M is a symmetric $$n \times n$$ matrix. A quadratic map from $$\mathbb{R}^n$$ to $$\mathbb{R}^m$$ is a map, all m of whose components are quadratic forms. One of the two central questions in this thesis is this: when is the image of a quadratic map $$Q: \mathbb{R}^n \rightarrow \mathbb{R}^m$$ a convex subset of $$\mathbb{R}^m$$? This question has intrinsic interest; despite being only a degree removed from linear maps, quadratic maps are not well understood. However, the convexity properties of quadratic maps have practical consequences as well: underlying every semidefinite program is a quadratic map, and the convexity of the image of that map determines the nature of the solutions to the semidefinite program. Quadratic maps that map into $$\mathbb{R}^2$$ and $$\mathbb{R}^3$$ have been studied before (in (Dines, 1940) and (Calabi, 1964) respectively). The Roundness Theorem, the first of the two principal results in this thesis, is a sufficient and (almost) necessary condition for a quadratic map $$Q: \mathbb{R}^n \rightarrow \mathbb{R}^m$$ to have a convex image when $$m \geq 4$$, $$n \geq m$$ and $$n \not= m + 1$$. Concomitant with the Roundness Theorem is an important lemma: when $$n < m$$, quadratic maps from $$\mathbb{R}^n$$ to $$\mathbb{R}^m$$seldom have convex images. This second result in this thesis is a controllability condition for bilinear systems defined on direct products of the form $$\mathcal{G} \times\mathcal{G}$$, where $$\mathcal{G}$$ is a simple Lie group. The condition is this: a bilinear system defined on $$\mathcal{G} \times\mathcal{G}$$ is not controllable if and only if the Lie algebra generated by the system’s vector fields is the graph of some automorphism of $$\mathcal{g}$$, the Lie algebra of $$\mathcal{G}$$. Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:11030574 Downloads of this work:

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