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Symmetry reduction in K − P problems

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##### Abstract

$K-P$ problems are a class of geometric optimal control problems on finite-dimensional real semisimple Lie groups which arise, for example, in the control of quantum systems when the Lie group is $SU(n)$. In these problems, the sub-Riemannian distribution corresponds to the $\p$-part of a Cartan decomposition (also known as the $-1$ eigenspace of a Cartan involution), and these systems are totally controllable. However, finding a particular optimal trajectory can be in general a computationally difficult problem, and from an analytic perspective, the Lie groups are sufficiently complicated to make finding such objects as the cut locus difficult. $K-P$ problems possess a natural symmetry arising from a conjugation action of the Lie group corresponding to the $\K$-part of the Cartan decomposition (also known as the Lie group corresponding to the $+1$ eigenspace of a Cartan involution). Then one may study the topological and geometric structure of the quotient space obtained under this action; however, this quotient space is no longer a manifold but a stratified space. On the regular part of this quotient space, conditions are given under which a Riemannian structure may be induced from the original sub-Riemannian structure. The optimal Riemannian geodesics on the quotient space which approach the equivalence class of ${\bf 1}$, the identity element of the Lie group in question, are shown to correspond exactly to optimal sub-Riemannian geodesics on the original space. When the Riemannian structure has certain properties, such as nonpositive sectional curvature, and when the regular part of the quotient space has certain topological features, such as simple connectedness, the cut locus of ${\bf 1}$ is shown to belong to the singular part of the stratified structure. The necessary condition to define this Riemannian structure is verified for all $K-P$ decompositions of $SU(n)$ and the cut locus of $SU(2)$ in particular is derived using the negative sectional curvature of the induced Riemannian structure. An explicit description of the quotient spaces for a particular class of $K-P$ decompositions of $SU(n)$ is described, showing that the regular part of these quotient spaces is simply connected. There is also some discussion of how one may be able to extend many of the proofs in this dissertation.