scholarly journals Boundary controllability for a coupled system of degenerate/singular parabolic equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Brahim Allal ◽  
Abdelkarim Hajjaj ◽  
Jawad Salhi ◽  
Amine Sbai
Keyword(s):  
Spectral Analysis ◽  
Parabolic Equations ◽  
Moment Method ◽  
Coupled System ◽  
Control Cost ◽  
Boundary Controllability ◽  
The Moment ◽  
Singular Parabolic Equations

<p style='text-indent:20px;'>In this paper we study the boundary controllability for a system of two coupled degenerate/singular parabolic equations with a control acting on only one equation. We analyze both approximate and null boundary controllability properties. Besides, we provide an estimate on the null-control cost. The proofs are based on a detailed spectral analysis and the use of the moment method by Fattorini and Russell together with some results on biorthogonal families.</p>

scholarly journals The cost of controlling strongly degenerate parabolic equations

2020 ◽  
Vol 26 ◽  
pp. 2
Author(s):  
P. Cannarsa ◽  
P. Martinez ◽  
J. Vancostenoble
Keyword(s):  
Parabolic Equations ◽  
Complex Analysis ◽  
Degenerate Parabolic Equations ◽  
Control Cost ◽  
Locally Distributed Control ◽  
Degenerate Parabolic ◽  
The Moment ◽  
The Cost ◽  
Strongly Degenerate ◽  
Degeneracy Parameter

We consider the typical one-dimensional strongly degenerate parabolic operator Pu = ut − (xαux)x with 0 < x < ℓ and α ∈ (0, 2), controlled either by a boundary control acting at x = ℓ, or by a locally distributed control. Our main goal is to study the dependence of the so-called controllability cost needed to drive an initial condition to rest with respect to the degeneracy parameter α. We prove that the control cost blows up with an explicit exponential rate, as eC/((2−α)2T), when α → 2− and/or T → 0+. Our analysis builds on earlier results and methods (based on functional analysis and complex analysis techniques) developed by several authors such as Fattorini-Russel, Seidman, Güichal, Tenenbaum-Tucsnak and Lissy for the classical heat equation. In particular, we use the moment method and related constructions of suitable biorthogonal families, as well as new fine properties of the Bessel functions Jν of large order ν (obtained by ordinary differential equations techniques).

scholarly journals A Boundary Estimate for Singular Sub-Critical Parabolic Equations

2020 ◽  
Author(s):  
Ugo Gianazza ◽  
Naian Liao
Keyword(s):  
Modulus Of Continuity ◽  
Weak Solutions ◽  
Parabolic Equations ◽  
Boundary Point ◽  
Cylindrical Domain ◽  
Boundary Estimate ◽  
Critical Range ◽  
Type Integral ◽  
Wiener Type ◽  
Singular Parabolic Equations

Abstract We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to singular parabolic equations of $p$-Laplacian type, with $p$ in the sub-critical range $\big(1,\frac{2N}{N+1}\big]$. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic $p$-capacity.

scholarly journals Spectral gap in random bipartite biregular graphs and applications

2021 ◽  
pp. 1-39
Author(s):  
Gerandy Brito ◽  
Ioana Dumitriu ◽  
Kameron Decker Harris
Keyword(s):  
Community Detection ◽  
Adjacency Matrix ◽  
Moment Method ◽  
Coding Theory ◽  
High Probability ◽  
Spectral Gap ◽  
Matrix Completion ◽  
Full Rank ◽  
The Moment ◽  
Deterministic Matrix

Abstract We prove an analogue of Alon’s spectral gap conjecture for random bipartite, biregular graphs. We use the Ihara–Bass formula to connect the non-backtracking spectrum to that of the adjacency matrix, employing the moment method to show there exists a spectral gap for the non-backtracking matrix. A by-product of our main theorem is that random rectangular zero-one matrices with fixed row and column sums are full rank with high probability. Finally, we illustrate applications to community detection, coding theory, and deterministic matrix completion.

Investigation of the Thermodynamic Properties of Anharmonic Crystals with Defects and Influence of Anharmonicity in Exafs by the Moment Method

1998 ◽  
Vol 12 (02) ◽  
pp. 191-205 ◽  
Author(s):  
Vu Van Hung ◽  
Nguyen Thanh Hai
Keyword(s):  
Thermal Expansion ◽  
Thermodynamic Properties ◽  
Phase Change ◽  
Specific Heat ◽  
Thermal Expansion Coefficient ◽  
Expansion Coefficient ◽  
Moment Method ◽  
Adiabatic Compressibility ◽  
Relative Displacement ◽  
The Moment

By the moment method established previously on the basis of the statistical mechanics, the thermodynamic properties of a strongly anharmonic face-centered and body-centered cubic crystal with point defect are considered. The thermal expansion coefficient, the specific heat Cv and Cp, the isothermal and adiabatic compressibility, etc. are calculated. Our calculated results of the thermal expansion coefficient, the specific heat Cv and Cp… of W, Nb, Au and Ag metals at various temperatures agrees well with the measured values. The anharmonic effects in extended X-ray absorption fine structure (EXAFS) in the single-shell model are considered. We have obtained a new formula for anharmonic contribution to the mean square relative displacement. The anharmonicity is proportional to the temperature and enters the phase change of EXAFS. Our calculated results of Debye–Waller factor and phase change in EXAFS of Cu at various temperatures agrees well with the measured values.

scholarly journals An Inverse Source Problem for Singular Parabolic Equations with Interior Degeneracy

2018 ◽  
Vol 2018 ◽  
pp. 1-16
Author(s):  
Khalid Atifi ◽  
Idriss Boutaayamou ◽  
Hamed Ould Sidi ◽  
Jawad Salhi
Keyword(s):  
Numerical Solution ◽  
Parabolic Equations ◽  
Measurement Data ◽  
Gradient Methods ◽  
Stability Estimate ◽  
Inverse Source Problem ◽  
Output Least Squares ◽  
Singular Parabolic Equations ◽  
Interior Degeneracy ◽  
Least Squares Approach

The main purpose of this work is to study an inverse source problem for degenerate/singular parabolic equations with degeneracy and singularity occurring in the interior of the spatial domain. Using Carleman estimates, we prove a Lipschitz stability estimate for the source term provided that additional measurement data are given on a suitable interior subdomain. For the numerical solution, the reconstruction is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. The Fréchet differentiability of the Tikhonov functional and the Lipschitz continuity of the Fréchet gradient are proved. These properties allow us to apply gradient methods for numerical solution of the considered inverse source problem.

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