43,254 research outputs found

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

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    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

    Information-theoretic approach to the study of control systems

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    We propose an information-theoretic framework for analyzing control systems based on the close relationship of controllers to communication channels. A communication channel takes an input state and transforms it into an output state. A controller, similarly, takes the initial state of a system to be controlled and transforms it into a target state. In this sense, a controller can be thought of as an actuation channel that acts on inputs to produce desired outputs. In this transformation process, two different control strategies can be adopted: (i) the controller applies an actuation dynamics that is independent of the state of the system to be controlled (open-loop control); or (ii) the controller enacts an actuation dynamics that is based on some information about the state of the controlled system (closed-loop control). Using this communication channel model of control, we provide necessary and sufficient conditions for a system to be perfectly controllable and perfectly observable in terms of information and entropy. In addition, we derive a quantitative trade-off between the amount of information gathered by a closed-loop controller and its relative performance advantage over an open-loop controller in stabilizing a system. This work supplements earlier results [H. Touchette, S. Lloyd, Phys. Rev. Lett. 84, 1156 (2000)] by providing new derivations of the advantage afforded by closed-loop control and by proposing an information-based optimality criterion for control systems. New applications of this approach pertaining to proportional controllers, and the control of chaotic maps are also presented.Comment: 18 pages, 7 eps figure

    The Nose-hoover thermostated Lorentz gas

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    We apply the Nose-Hoover thermostat and three variations of it, which control different combinations of velocity moments, to the periodic Lorentz gas. Switching on an external electric field leads to nonequilibrium steady states for the four models with a constant average kinetic energy of the moving particle. We study the probability density, the conductivity and the attractor in nonequilibrium and compare the results to the Gaussian thermostated Lorentz gas and to the Lorentz gas as thermostated by deterministic scattering.Comment: 7 pages (revtex) with 10 figures (postscript), most of the figures are bitmapped with low-resolution. The originals are many MB, they can be obtained upon reques

    A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws

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    In this article we consider one-dimensional random systems of hyperbolic conservation laws. We first establish existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws which involve random initial data and random flux functions. Based on these results we present an a posteriori error analysis for a numerical approximation of the random entropy admissible solution. For the stochastic discretization, we consider a non-intrusive approach, the Stochastic Collocation method. The spatio-temporal discretization relies on the Runge--Kutta Discontinuous Galerkin method. We derive the a posteriori estimator using continuous reconstructions of the discrete solution. Combined with the relative entropy stability framework this yields computable error bounds for the entire space-stochastic discretization error. The estimator admits a splitting into a stochastic and a deterministic (space-time) part, allowing for a novel residual-based space-stochastic adaptive mesh refinement algorithm. We conclude with various numerical examples investigating the scaling properties of the residuals and illustrating the efficiency of the proposed adaptive algorithm

    From Knowledge, Knowability and the Search for Objective Randomness to a New Vision of Complexity

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    Herein we consider various concepts of entropy as measures of the complexity of phenomena and in so doing encounter a fundamental problem in physics that affects how we understand the nature of reality. In essence the difficulty has to do with our understanding of randomness, irreversibility and unpredictability using physical theory, and these in turn undermine our certainty regarding what we can and what we cannot know about complex phenomena in general. The sources of complexity examined herein appear to be channels for the amplification of naturally occurring randomness in the physical world. Our analysis suggests that when the conditions for the renormalization group apply, this spontaneous randomness, which is not a reflection of our limited knowledge, but a genuine property of nature, does not realize the conventional thermodynamic state, and a new condition, intermediate between the dynamic and the thermodynamic state, emerges. We argue that with this vision of complexity, life, which with ordinary statistical mechanics seems to be foreign to physics, becomes a natural consequence of dynamical processes.Comment: Phylosophica
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