On the approximation of a small solution from a forward-backward equation

2013 ◽  
Author(s):  
M. F. Teodoro
Keyword(s):  
Small Solution ◽  
Backward Equation

Embedding non-linear filtering in autonomous underwater vehicle dynamics via the Kolmogorov backward equation

2021 ◽  
pp. 014233122110196
Author(s):  
Ravish H. Hirpara ◽  
Shambhu N. Sharma
Keyword(s):  
State Vector ◽  
Autonomous Underwater Vehicle ◽  
Underwater Vehicle ◽  
Linear Filtering ◽  
Homogeneous Markov Process ◽  
Non Linear ◽  
Backward Equation ◽  
Noise Correction ◽  
Ito Process ◽  
Correction Terms

This paper revisits the state vector of an autonomous underwater vehicle (AUV) dynamics coupled with the underwater Markovian stochasticity in the ‘non-linear filtering’ context. The underwater stochasticity is attributed to atmospheric turbulence, planetary interactions, sea surface conditions and astronomical phenomena. In this paper, we adopt the Itô process, a homogeneous Markov process, to describe the AUV state vector evolution equation. This paper accounts for the process noise as well as observation noise correction terms by considering the underwater filtering model. The non-linear filtering of the paper is achieved using the Kolmogorov backward equation and the evolution of the conditional characteristic function. The non-linear filtering equation is the cornerstone formalism of stochastic optimal control systems. Most notably, this paper introduces the non-linear filtering theory into an underwater vehicle stochastic system by constructing a lemma and a theorem for the underwater vehicle stochastic differential equation that were not available in the literature.

scholarly journals The Concepts of Well-Posedness and Stability in Different Function Spaces for the 1D Linearized Euler Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Stefan Balint ◽  
Agneta M. Balint
Keyword(s):  
Euler Equation ◽  
Function Space ◽  
Euler Equations ◽  
Small Perturbation ◽  
Function Spaces ◽  
Well Posedness ◽  
Small Solution ◽  
Null Solution ◽  
Linearized Euler Equations ◽  
The Stability

This paper considers the stability of constant solutions to the 1D Euler equation. The idea is to investigate the effect of different function spaces on the well-posedness and stability of the null solution of the 1D linearized Euler equations. It is shown that the mathematical tools and results depend on the meaning of the concepts “perturbation,” “small perturbation,” “solution of the propagation problem,” and “small solution, that is, solution close to zero,” which are specific for each function space.

scholarly journals Verifying a Solver for Linear Mixed Integer Arithmetic in Isabelle/HOL

2020 ◽  
pp. 233-250
Author(s):  
Ralph Bottesch ◽  
Max W. Haslbeck ◽  
Alban Reynaud ◽  
René Thiemann
Keyword(s):  
Linear Programming ◽  
Branch And Bound ◽  
Decision Procedure ◽  
Simplex Algorithm ◽  
Upper Bounds ◽  
Mixed Integer ◽  
Small Solution ◽  
Code Generator ◽  
Integer Arithmetic ◽  
Termination Proofs

AbstractWe implement a decision procedure for linear mixed integer arithmetic and formally verify its soundness in Isabelle/HOL. We further integrate this procedure into one application, namely into , a formally verified certifier to check untrusted termination proofs. This checking involves assertions of unsatisfiability of linear integer inequalities; previously, only a sufficient criterion for such checks was supported. To verify the soundness of the decision procedure, we first formalize the proof that every satisfiable set of linear integer inequalities also has a small solution, and give explicit upper bounds. To this end we mechanize several important theorems on linear programming, including statements on integrality and bounds. The procedure itself is then implemented as a branch-and-bound algorithm, and is available in several languages via Isabelle’s code generator. It internally relies upon an adapted version of an existing verified incremental simplex algorithm.

Big Problem-Small Solution: Nanotechnology-Based Sealing Fluid

2018 ◽  
Author(s):  
Larry Todd ◽  
Matthew Cleveland ◽  
Kevin Docherty ◽  
James Reid ◽  
Kenneth Cowan ◽  
...  
Keyword(s):  
Small Solution

The Effects of Cathodic Reagent Concentration and Small Solution Volumes on the Corrosion of Copper in Dilute Nitric Acid Solutions

CORROSION ◽  
10.5006/2655 ◽  
2017 ◽  
Vol 74 (3) ◽  
pp. 326-336 ◽  
Author(s):  
J. Turnbull ◽  
R. Szukalo ◽  
M. Behazin ◽  
D. Hall ◽  
D. Zagidulin ◽  
...  
Keyword(s):  
Nitric Acid ◽  
Cross Sections ◽  
Focused Ion Beam ◽  
Ion Beam ◽  
Exposure Period ◽  
Geologic Repository ◽  
Small Solution ◽  
High Level Nuclear Waste ◽  
Nitrate And Nitrite ◽  
High Level

The exposure conditions experienced by copper-coated high-level nuclear waste containers in a deep geologic repository will evolve with time. An early exposure period involving the gamma irradiation of aerated humid vapor could lead to the formation of nitric acid condensed in limited volumes of water on the container surface. The evolution of the corrosion processes under these conditions have been studied using pH measurements in limited volumes of water containing various concentrations of nitric acid. The extent and morphology of corrosion was examined using scanning electron microscopy on surfaces and on focused ion beam cut cross sections. The composition of corrosion products was determined by energy dispersive x-ray analyses and Raman spectroscopy. In the absence of dissolved oxygen only minor corrosion was observed with the reduction of nitric acid inhibited by the formation of either chemisorbed nitrate and nitrite species or the formation of a thin cuprite (Cu2O) layer. When the solution was aerated, both oxygen and nitric acid acted as cathodic reagents. After extensive exposure periods corrosion was stifled by the formation of corrosion product deposits of Cu2O, CuO (tenorite), and Cu2NO3(OH)3 (rouaite).

scholarly journals The infinitesimal generator of the stochastic Burgers equation

2020 ◽  
Vol 178 (3-4) ◽  
pp. 1067-1124
Author(s):  
Massimiliano Gubinelli ◽  
Nicolas Perkowski
Keyword(s):  
Burgers Equation ◽  
Martingale Problem ◽  
Uniqueness Result ◽  
Stochastic Pdes ◽  
Martingale Approach ◽  
Infinite Dimensional ◽  
Cylinder Test ◽  
Stochastic Burgers Equation ◽  
Trivial Intersection ◽  
Backward Equation

Abstract We develop a martingale approach for a class of singular stochastic PDEs of Burgers type (including fractional and multi-component Burgers equations) by constructing a domain for their infinitesimal generators. It was known that the domain must have trivial intersection with the usual cylinder test functions, and to overcome this difficulty we import some ideas from paracontrolled distributions to an infinite dimensional setting in order to construct a domain of controlled functions. Using the new domain, we are able to prove existence and uniqueness for the Kolmogorov backward equation and the martingale problem. We also extend the uniqueness result for “energy solutions” of the stochastic Burgers equation of Gubinelli and Perkowski (J Am Math Soc 31(2):427–471, 2018) to a wider class of equations. As applications of our approach we prove that the stochastic Burgers equation on the torus is exponentially $$L^2$$ L 2 -ergodic, and that the stochastic Burgers equation on the real line is ergodic.

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