The Convexity of Quadratic Maps and the Controllability of Coupled Systems

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}).