Capacity of Control for Stochastic Dynamical Systems Perturbed by Mixed Fractional Brownian Motion with Delay in Control

In this paper, we discuss the relationships between capacity of control in entropy theory and intrinsic properties in control theory for a class of finite dimensional stochastic dynamical systems described by a linear stochastic differential equations driven by mixed fractional Brownian motion with delay in control. Stochastic dynamical systems can be described as an information channel between the space of control signals and the state space. We study this control to state information capacity of this channel in continuous time. We turned out that, the capacity of control depends on the time of final state in dynamical systems. By using the analysis and representation of fractional Gaussian process, the closed form of continuous optimal control law is derived. The reached optimal control law maximizes the mutual information between control signals and future state over a finite time horizon. The results obtained here are motivated by control to state information capacity for linear systems in both types deterministic and stochastic models that are widely used to understand information flows in wireless network information theory. The contribution of this paper is that we propose some new relationships between control theory and entropy theoretic properties of stochastic dynamical systems with delay in control. Finally, we present an example that serve to illustrate the relationships between capacity of control and intrinsic properties in control theory.Comment: 17 pages, 2 example

Nonsemimartingales: Stochastic differential equations and weak Dirichlet processes

In this paper we discuss existence and uniqueness for a one-dimensional time inhomogeneous stochastic differential equation directed by an $\mathbb{F}$-semimartingale $M$ and a finite cubic variation process $\xi$ which has the structure $Q+R$, where $Q$ is a finite quadratic variation process and $R$ is strongly predictable in some technical sense: that condition implies, in particular, that $R$ is weak Dirichlet, and it is fulfilled, for instance, when $R$ is independent of $M$. The method is based on a transformation which reduces the diffusion coefficient multiplying $\xi$ to 1. We use generalized It\^{o} and It\^{o}--Wentzell type formulae. A similar method allows us to discuss existence and uniqueness theorem when $\xi$ is a H\"{o}lder continuous process and $\sigma$ is only H\"{o}lder in space. Using an It\^{o} formula for reversible semimartingales, we also show existence of a solution when $\xi$ is a Brownian motion and $\sigma$ is only continuous.Comment: Published at http://dx.doi.org/10.1214/009117906000000566 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Organization of the magnetosphere during substorms

The change in degree of organization of the magnetosphere during substorms is investigated by analyzing various geomagnetic indices, as well as interplanetary magnetic field z-component and solar wind flow speed. We conclude that the magnetosphere self-organizes globally during substorms, but neither the magnetosphere nor the solar wind become more predictable in the course of a substorm. This conclusion is based on analysis of five hundred substorms in the period from 2000 to 2002. A minimal dynamic-stochastic model of the driven magnetosphere that reproduces many statistical features of substorm indices is discussed