Full Information H2 Control of Borel-Measurable Markov Jump Systems with Multiplicative Noises

This paper addresses an H2 optimal control problem for a class of discrete-time stochastic systems with Markov jump parameter and multiplicative noises. The involved Markov jump parameter is a uniform ergodic Markov chain taking values in a Borel-measurable set. In the presence of exogenous white noise disturbance, Gramian characterization is derived for the H2 norm, which quantifies the stationary variance of output response for the considered systems. Moreover, under the condition that full information of the system state is accessible to measurement, an H2 dynamic optimal control problem is shown to be solved by a zero-order stabilizing feedback controller, which can be represented in terms of the stabilizing solution to a set of coupled stochastic algebraic Riccati equations. Finally, an iterative algorithm is provided to get the approximate solution of the obtained Riccati equations, and a numerical example illustrates the effectiveness of the proposed algorithm.

Origins of Scaling Laws in Microbial Dynamics

Analysis of high-resolution time series data from the human and mouse gut microbiomes revealed that the gut microbial dynamics can be characterized by several robust and simple scaling laws. It is still unknown if those scaling laws are universal across different body sites, host species, or even free-living microbial communities. Moreover, the underlying mechanisms responsible for those scaling laws remain poorly understood. Here, we demonstrate that those scaling laws are not unique to gut microbiome, but universal across different habitats, from human skin and oral microbiome to marine plankton bacteria and eukarya communities. Since completely shuffled time series yield very similar scaling laws, we conjecture that the universal scaling laws in various microbiomes are largely driven by temporal stochasticity of the host or environmental factors. We leverage a simple population dynamics model with both deterministic inter-species interactions and stochastic noise to confirm our conjecture. In particular, we find that those scaling laws are jointly determined by inter-species interactions and linear multiplicative noises. The presented results deepen our understanding of the nature of scaling laws in microbial dynamics.