scholarly journals Asymptotic Hölder regularity for the ellipsoid process

2020 ◽  
Vol 26 ◽  
pp. 112 ◽  
Author(s):  
Ángel Arroyo ◽  
Mikko Parviainen
Keyword(s):  
Stochastic Process ◽  
Dynamic Programming ◽  
Random Walk ◽  
Elliptic Equations ◽  
Divergence Form ◽  
Dynamic Programming Principle ◽  
Step Size ◽  
Measurable Coefficients ◽  
Continuity Assumption ◽  
Hölder Estimate

We obtain an asymptotic Hölder estimate for functions satisfying a dynamic programming principle arising from a so-called ellipsoid process. By the ellipsoid process we mean a generalization of the random walk where the next step in the process is taken inside a given space dependent ellipsoid. This stochastic process is related to elliptic equations in non-divergence form with bounded and measurable coefficients, and the regularity estimate is stable as the step size of the process converges to zero. The proof, which requires certain control on the distortion and the measure of the ellipsoids but not continuity assumption, is based on the coupling method.

scholarly journals A sharp Hölder estimate for elliptic equations in two variables

2005 ◽  
Vol 135 (1) ◽  
pp. 165-173 ◽  
Author(s):  
Tonia Ricciardi
Keyword(s):  
Elliptic Equations ◽  
Type Inequality ◽  
Divergence Form ◽  
Two Dimensional ◽  
Measurable Coefficients ◽  
The Matrix ◽  
Hölder Estimate

We prove a sharp Hölder estimate for solutions of linear, two-dimensional, divergence-form, elliptic equations with measurable coefficients, such that the matrix of the coefficients is symmetric and has unit determinant. Our result extends some previous work by Piccinini and Spagnolo. The proof relies on a sharp Wirtinger type inequality.

Nonlinear Stochastic Optimal Control of MDOF Partially Observable Linear Systems Excited by Combined Harmonic and Wide-Band Noises

2019 ◽  
Vol 19 (03) ◽  
pp. 1950019 ◽  
Author(s):  
R. C. Hu ◽  
X. F. Wang ◽  
X. D. Gu ◽  
R. H. Huan
Keyword(s):  
Optimal Control ◽  
Dynamic Programming ◽  
Control Problem ◽  
Linear Systems ◽  
Control Strategy ◽  
Stochastic Optimal Control ◽  
Wide Band ◽  
Dynamic Programming Principle ◽  
Stochastic Dynamic ◽  
Partially Observable

In this paper, nonlinear stochastic optimal control of multi-degree-of-freedom (MDOF) partially observable linear systems subjected to combined harmonic and wide-band random excitations is investigated. Based on the separation principle, the control problem of a partially observable system is converted into a completely observable one. The dynamic programming equation for the completely observable control problem is then set up based on the stochastic averaging method and stochastic dynamic programming principle, from which the nonlinear optimal control law is derived. To illustrate the feasibility and efficiency of the proposed control strategy, the responses of the uncontrolled and optimal controlled systems are respectively obtained by solving the associated Fokker–Planck–Kolmogorov (FPK) equation. Numerical results show the proposed control strategy can dramatically reduce the response of stochastic systems subjected to both harmonic and wide-band random excitations.

scholarly journals A Swarm Robotic Exploration Strategy Based on an Improved Random Walk Method

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Bao Pang ◽  
Yong Song ◽  
Chengjin Zhang ◽  
Hongling Wang ◽  
Runtao Yang
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Swarm Robotics ◽  
Search Strategy ◽  
Lévy Flight ◽  
Step Size ◽  
Levy Flight ◽  
Random Walk Method ◽  
Robotic Exploration ◽  
Exploration Mission

An environment can be searched far more efficiently if the appropriate search strategy is used. Because of the limited individual abilities of swarm robots, namely, local sensing and low processing power, random searching is the main search strategy used in swarm robotics. The random walk methods that are used most commonly are Brownian motion and Lévy flight, both of which mimic the self-organized behavior of social insects. However, both methods are somewhat limited when applied to swarm robotics, where having the robots search repeatedly can result in highly inefficient searching. Therefore, by analyzing the characteristics of swarm robotic exploration, this paper proposes an improved random walk method in which each robot adjusts its step size adaptively to reduce the number of repeated searches by estimating the density of robots in the environment. Simulation experiments and experiments with actual robots are conducted to study the effectiveness of the proposed method and evaluate its performance in an exploration mission. The experimental results presented in this paper show that an area is covered more efficiently using the proposed method than it is using either Brownian motion or Lévy flight.

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