No abstract

Abstract. Many processes in cell biology encode and process information and enact responses by modulating the concentrations of biological molecules. Such modulations serve functions ranging from encoding and transmitting information about external stimuli to regulating internal metabolic states. To understand how such processes operate requires gaining insights into the basic mechanisms by which biochemical species interact and respond to internal and external perturbations. One approach is to model the biochemical species concentrations through the van Kampen Linear Noise Equations, which account for the change in biochemical concentrations from reactions and account for fluctuations in concentrations. For many systems, the Linear Noise Equations exhibit stiffness as a consequence of the chemical reactions occurring at significantly different rates. This presents challenges in the analysis of the kinetics and in performing efficient numerical simulations. To deal with this source of stiffness and to obtain reduced models more amenable to analysis, we present a systematic procedure for obtaining effective stochastic dynamics for the chemical species having relatively slow characteristic time scales while eliminating representations of the chemical species having relatively fast characteristic time scales. To demonstrate the applicability of this multiscale technique in the context of Linear Noise Equations, the reduction is applied to models of gene regulatory networks. Results are presented which compare numerical results for the full system to the reduced descriptions. The presented stochastic reduction procedure provides a potentially versatile tool for systematically obtaining reduced approximations of Linear Noise Equations.

This paper addresses a detection problem where several spatially distributed sensors independently observe a time-inhomogeneous stochastic process. The task is to decide between two hypotheses regarding the statistics of the observed process at the end of a fixed time interval. In the proposed method, each of the sensors transmits once to a fusion center a locally processed summary of its information in the form of a likelihood ratio. The fusion center then combines these messages to arrive at an optimal decision in the Neyman-Pearson framework. The approach is motivated by applications arising in the detection of mobile radioactive sources, and offers a pathway toward the development of novel fixedinterval detection algorithms that combine decentralized processing with optimal centralized decision making.

This paper presents a receding horizon control design for a robot subject to stochastic uncertainty, moving in a constrained environment. Instead of minimizing the expectation of a cost functional while ensuring satisfaction of probabilistic state constraints, we propose a two-stage solution where the path that minimizes the cost functional is planned deterministically, and a local stochastic optimal controller with exit constraints ensures satisfaction of probabilistic state constraints while following the planned path. This control design strategy ensures boundedness of errors around the reference path and collision-free convergence to the goal with probability one under the assumption of unbounded inputs. We show that explicit expressions for the control law are possible for certain cases. We provide simulation results for a point robot moving in a constrained two-dimensional environment under Brownian noise. The method can be extended to systems with bounded inputs, if a small nonzero probability of failure can be accepted.

This paper addresses a detection problem where several spatially distributed sensors independently observe a time-inhomogeneous stochastic process. The task is to decide at the end of a fixed time interval between two hypotheses regarding the statistics of the observed process. In the proposed method, each of the sensors transmits once to a fusion center a locally processed summary of its information in the form of a likelihood ratio. The fusion center then combines these messages to arrive at an optimal decision in the Neyman-Pearson framework. The approach is motivated by applications arising in the detection of mobile radioactive sources, and it serves as a first step toward the development of novel fixed-interval detection algorithms that combine decentralized processing with optimal centralized decision making.

<p style='text-indent:20px;'>In this paper, we study the dynamics of a linear control system with given state feedback control law in the presence of fast periodic sampling at temporal frequency <inline-formula><tex-math id="M1">\begin{document}$ 1/\delta $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M2">\begin{document}$ 0 &lt; \delta \ll 1 $\end{document}</tex-math></inline-formula>), together with small white noise perturbations of size <inline-formula><tex-math id="M3">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M4">\begin{document}$ 0&lt; \varepsilon \ll 1 $\end{document}</tex-math></inline-formula>) in the state dynamics. For the ensuing continuous-time stochastic process indexed by two small parameters <inline-formula><tex-math id="M5">\begin{document}$ \varepsilon,\delta $\end{document}</tex-math></inline-formula>, we obtain effective ordinary and stochastic differential equations describing the mean behavior and the typical fluctuations about the mean in the limit as <inline-formula><tex-math id="M6">\begin{document}$ \varepsilon,\delta \searrow 0 $\end{document}</tex-math></inline-formula>. The effective fluctuation process is found to vary, depending on whether <inline-formula><tex-math id="M7">\begin{document}$ \delta \searrow 0 $\end{document}</tex-math></inline-formula> faster than/at the same rate as/slower than <inline-formula><tex-math id="M8">\begin{document}$ \varepsilon \searrow 0 $\end{document}</tex-math></inline-formula>. The most interesting case is found to be the one where <inline-formula><tex-math id="M9">\begin{document}$ \delta, \varepsilon $\end{document}</tex-math></inline-formula> are comparable in size; here, the limiting stochastic differential equation for the fluctuations has both a diffusive term due to the small noise and an effective drift term which captures the cumulative effect of the fast sampling. In this regime, our results yield a time-inhomogeneous Markov process which provides a strong (pathwise) approximation of the original non-Markovian process, together with estimates on the ensuing error. A simple example involving an infinite time horizon linear quadratic regulation problem illustrates the results.</p>

No abstract

We consider a nonlinear differential equation under the combined influence of small statedependent Brownian perturbations of size ε, and fast periodic sampling with period δ; 0 < ε, δ 1. Thus, state samples (measurements) are taken every δ time units, and the instantaneous rate of change of the state depends on its current value as well as its most recent sample. We show that the resulting stochastic process indexed by ε, δ, can be approximated, as ε, δ 0, by an ordinary differential equation (ode) with vector field obtained by replacing the most recent sample by the current value of the state.We next analyze the fluctuations of the stochastic process about the limiting ode. Our main result asserts that, for the case when δ 0 at the same rate as, or faster than, ε 0, the rescaled fluctuations can be approximated in a suitable strong (pathwise) sense by a limiting stochastic differential equation (sde). This sde varies depending on the exact rates at which ε, δ 0. The key contribution here involves computing the effective drift term capturing the interplay between noise and sampling in the limiting sde. The results essentially provide a first-order perturbation expansion, together with error estimates, for the stochastic process of interest. Connections with the performance analysis of feedback control systems with sampling are discussed and illustrated numerically through a simple example.