scholarly journals Numerical null controllability of the heat equation through a least squares and variational approach

2014 ◽  
Vol 25 (3) ◽  
pp. 277-306 ◽  
Author(s):  
ARNAUD MÜNCH ◽  
PABLO PEDREGAL
Keyword(s):  
Heat Equation ◽  
Variational Approach ◽  
A Priori ◽  
Conjugate Function ◽  
Adjoint System ◽  
Null Controllability ◽  
Initial State ◽  
Recent Developments ◽  
Minimization Algorithms ◽  
The One

This work is concerned with the numerical computation of null controls for the heat equation. The goal is to compute an approximation of controls that drives the solution from a prescribed initial state at t=0 to zero at t=T. In spite of the diffusion of the heat equation, recent developments indicate that this issue is difficult and still largely open. Most of the existing literature, concerned with controls of minimal L2-norm, make use of dual convex arguments and introduce backward adjoint system. In practice, the null control problem is then reduced to the minimization of a dual conjugate function with respect to the final condition of the adjoint state. As a consequence of the highly regularizing property of the heat kernel, this final condition – which may be seen as the Lagrange multiplier for the null controllability condition – does not belong to L2, but to a much larger space than can hardly be approximated by finite (discrete) dimensional basis. This phenomenon, unavoidable whatever be the numerical approximation used, strongly deteriorates the efficiency of minimization algorithms. In this work, we do not use duality arguments and in particular do not introduce any backward heat equation. For the boundary case, the approach consists first in introducing a class of functions satisfying a priori the boundary conditions in space and time, in particular the null controllability condition at time T, and then finding among this class one element satisfying the heat equation. This second step is done by minimizing a convex functional among the admissible corrector functions of the heat equation. The inner case is performed in a similar way. We present the (variational) approach, discuss the main features of it and then describe some numerical experiments highlighting the interest of the method. The method holds in any dimension but, for the sake of simplicity, we provide details in the one-space dimensional case.

scholarly journals Method of External Potential in Solution of Cauchy Mixed Problem for the Heat Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
T. Sh. Kalmenov ◽  
N. E. Tokmagambetov
Keyword(s):  
Heat Equation ◽  
Mixed Problem ◽  
A Priori Estimates ◽  
A Priori ◽  
Practical Importance ◽  
Fourier Method ◽  
Integral Form ◽  
Heat Equations ◽  
Main Research ◽  
The One

Numerous research works are devoted to study Cauchy mixed problem for model heat equations because of its theoretical and practical importance. Among them we can notice monographers Vladimirov (1988), Ladyzhenskaya (1973), and Tikhonov and Samarskyi (1980) which demonstrate main research methods, such as Fourier method, integral equations method, and the method of a priori estimates. But at the same time to represent the solution of Cauchy mixed problem in integral form by given and known functions has not been achieved up to now. This paper completes this omission for the one-dimensional heat equation.

scholarly journals NULL CONTROLLABILITY OF DEGENERATE NONAUTONOMOUS PARABOLIC EQUATIONS

2019 ◽  
pp. 311
Author(s):  
Abbes Benaissa ◽  
Abdelatif Kainane Mezadek ◽  
Lahcen Maniar
Keyword(s):  
Parabolic Equation ◽  
Open Subset ◽  
Parabolic Equations ◽  
Diffusion Coefficients ◽  
Adjoint System ◽  
Null Controllability ◽  
Carleman Estimates ◽  
One Dimensional ◽  
Neumann Type ◽  
The One

In this paper we are interested in the study of the null controllability for the one dimensional degenerate non autonomous parabolic equation$$u_{t}-M(t)(a(x)u_{x})_{x}=h\chi_{\omega},\qquad  (x,t)\in Q=(0,1)\times(0,T),$$ where $\omega=(x_{1},x_{2})$ is asmall nonempty open subset in $(0,1)$, $h\in L^{2}(\omega\times(0,T))$, the diffusion coefficients $a(\cdot)$ isdegenerate at $x=0$ and $M(\cdot)$ is non degenerate on $[0,T]$. Also the boundary conditions are considered tobe Dirichlet or Neumann type related to the degeneracy rate of $a(\cdot)$. Under some conditions on the functions$a(\cdot)$ and $M(\cdot)$, we prove some global Carleman estimates which will yield the  observability inequalityof the associated adjoint system and equivalently the null controllability of our parabolic equation.

ON EXACT DEAD-CORE RATES FOR A SEMILINEAR HEAT EQUATION WITH STRONG ABSORPTION

2011 ◽  
Vol 13 (01) ◽  
pp. 1-52 ◽  
Author(s):  
YUKIHIRO SEKI
Keyword(s):  
Heat Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Strong Absorption ◽  
Matched Asymptotic Expansion ◽  
Semilinear Heat Equation ◽  
Dead Core ◽  
Priori Estimates ◽  
Expansion Technique ◽  
The One

We study a semilinear heat equation with strong absorption ut = Δu - up with 0 < p < 1 in RN. A solution is known to develop dead-core in finite time for a wide class of initial data. We construct specific solutions with exact dead-core rates faster than the one given by the corresponding ODE. They are constructed formally by means of matched asymptotic expansion technique and rigorously by means of topological fixed-point arguments based on a priori estimates. To obtain the a priori estimates, we analyze a certain linearized problem in a new function space [Formula: see text].

Biosecurity: tools, behaviours and concepts

2020 ◽  
Vol 4 (5) ◽  
pp. 449-452
Author(s):  
Alan MacLeod ◽  
Nicola Spence
Keyword(s):  
Plant Protection ◽  
Human Life ◽  
Animal Health ◽  
Early Warning Systems ◽  
Policy Makers ◽  
Recent Developments ◽  
Surveillance Programs ◽  
Harmful Organisms ◽  
The One ◽  
Biosecurity Practices

COVID 19 has raised the profile of biosecurity. However, biosecurity is not only about protecting human life. This issue brings together mini-reviews examining recent developments and thinking around some of the tools, behaviours and concepts around biosecurity. They illustrate the multi-disciplinary nature of the subject, demonstrating the interface between research and policy. Biosecurity practices aim to prevent the spread of harmful organisms; recognising that 2020 is the International Year of Plant Health, several focus on plant biosecurity although invasive species and animal health concerns are also captured. The reviews show progress in developing early warning systems and that plant protection organisations are increasingly using tools that compare multiple pest threats to prioritise responses. The bespoke modelling of threats can inform risk management responses and synergies between meteorology and biosecurity provide opportunities for increased collaboration. There is scope to develop more generic models, increasing their accessibility to policy makers. Recent research can improve pest surveillance programs accounting for real-world constraints. Social science examining individual farmer behaviours has informed biosecurity policy; taking a broader socio-cultural approach to better understand farming networks has the potential to change behaviours in a new way. When encouraging public recreationists to adopt positive biosecurity behaviours communications must align with their values. Bringing together the human, animal, plant and environmental health sectors to address biosecurity risks in a common and systematic manner within the One Biosecurity concept can be achieved through multi-disciplinary working involving the life, physical and social sciences with the support of legislative bodies and the public.

scholarly journals Minkowski box from Yangian bootstrap

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Luke Corcoran ◽  
Florian Loebbert ◽  
Julian Miczajka ◽  
Matthias Staudacher
Keyword(s):  
Minkowski Space ◽  
Wigner Function ◽  
Functional Form ◽  
A Priori ◽  
Alternative Methods ◽  
Feynman Integrals ◽  
The One

Abstract We extend the recently developed Yangian bootstrap for Feynman integrals to Minkowski space, focusing on the case of the one-loop box integral. The space of Yangian invariants is spanned by the Bloch-Wigner function and its discontinuities. Using only input from symmetries, we constrain the functional form of the box integral in all 64 kinematic regions up to twelve (out of a priori 256) undetermined constants. These need to be fixed by other means. We do this explicitly, employing two alternative methods. This results in a novel compact formula for the box integral valid in all kinematic regions of Minkowski space.

scholarly journals Supergraph calculation of one-loop divergences in higher-derivative 6D SYM theory

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
I. L. Buchbinder ◽  
E. A. Ivanov ◽  
B. S. Merzlikin ◽  
K. V. Stepanyantz
Keyword(s):  
Effective Action ◽  
A Priori ◽  
Coupling Constant ◽  
Harmonic Superspace ◽  
Classical Action ◽  
Higher Derivative ◽  
Component Approach ◽  
Dimensionless Coupling ◽  
The One ◽  
Dimensionless Coupling Constant

Abstract We apply the harmonic superspace approach for calculating the divergent part of the one-loop effective action of renormalizable 6D, $$ \mathcal{N} $$ N = (1, 0) supersymmetric higher-derivative gauge theory with a dimensionless coupling constant. Our consideration uses the background superfield method allowing to carry out the analysis of the effective action in a manifestly gauge covariant and $$ \mathcal{N} $$ N = (1, 0) supersymmetric way. We exploit the regularization by dimensional reduction, in which the divergences are absorbed into a renormalization of the coupling constant. Having the expression for the one-loop divergences, we calculate the relevant β-function. Its sign is specified by the overall sign of the classical action which in higher-derivative theories is not fixed a priori. The result agrees with the earlier calculations in the component approach. The superfield calculation is simpler and provides possibilities for various generalizations.

scholarly journals A Business Model Framework to Characterize Digital Multisided Platforms

2020 ◽  
Vol 6 (1) ◽  
pp. 10 ◽  
Author(s):  
Marco Ardolino ◽  
Nicola Saccani ◽  
Federico Adrodegari ◽  
Marco Perona
Keyword(s):  
Information And Communication Technologies ◽  
Assessment Tool ◽  
Empirical Studies ◽  
Communication Technologies ◽  
Model Framework ◽  
Recent Developments ◽  
Information And Communication ◽  
Multiple Case ◽  
The One

Businesses grounded upon multisided platforms (MSPs) are found in a growing number of industries, thanks to the recent developments in Internet and digital technologies. Digital MSPs enable multiple interactions among users of different sides through information and communication technologies. The understanding of the characteristics and constituents of MSPs is fragmented along different literature streams. Moreover, very few empirical studies have been carried out to date. In order to fill this gap, this paper presents a three-level framework that describes a digital MSP. The proposed framework is based on literature analysis and multiple case study. On the one hand, the framework can be used to describe MSP as it provides an operationalization of the concept through the identification of specific dimensions, variables and items; on the other hand, it can be used as an assessment tool by practitioners, as exemplified by the three empirical applications presented in this paper.

Sign in / Sign up

Export Citation Format

Share Document