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In this paper, we consider direct policy optimization for the linear-quadratic Gaussian (LQG) setting. Over the past few years, it has been recognized that the landscape of stabilizing output-feedback controllers of relevance to LQG has an intricate geometry, particularly as it pertains to the existence of spurious stationary points. In order to address such challenges, in this paper, we first adopt a Riemannian metric for the space of stabilizing full-order minimal output-feedback controllers. We then proceed to prove that the orbit space of such controllers modulo coordinate transformation admits a Riemannian quotient manifold structure. This geometric structure is then used to develop a Riemannian gradient descent for the direct LQG policy optimization. We prove a local convergence guarantee with linear rate and show the proposed approach exhibits significantly faster and more robust numerical performance as compared with ordinary gradient descent for LQG. Subsequently, we provide reasons for this observed behavior; in particular, we argue that optimizing over the orbit space of controllers is the right theoretical and computational setup for direct LQG policy optimization. 

We consider the propagation of light in a random medium of two-level atoms. We investigate the dynamics of the field and atomic probability amplitudes for a two-photon state and show that at long times and large distances, the corresponding average probability densities can be determined from the solutions to a system of kinetic equations. 

In this paper, we develop a distributed consensus algorithm for agents whose states evolve on a manifold. This algorithm is complementary to traditional consensus, predominantly developed for systems with dynamics on vector spaces. We provide theoretical convergence guarantees for the proposed manifold consensus provided that agents are initialized within a geodesically convex (g-convex) set. This required condition on initialization is not restrictive as g-convex sets may be comparatively “large” for relevant Riemannian manifolds. Our approach to manifold consensus builds upon the notion of Riemannian Center of Mass (RCM) and the intrinsic structure of the manifold to avoid projections in the ambient space. We first show that on a g-convex ball, all states coincide if and only if each agent’s state is the RCM of its neighbors’ states. This observation facilitates our convergence guarantee to the consensus submanifold. Finally, we provide simulation results that exemplify the linear convergence rate of the proposed algorithm and illustrates its statistical properties over randomly generated problem instances. 

We consider the theory of spontaneous emission for a random medium of stationary two-level atoms. We investigate the dynamics of the field and atomic probability amplitudes for a one-photon state of the system. At long times and large distances, we show that the corresponding average probability densities can be determined from the solutions to a pair of kinetic equations.