scholarly journals Ill-conditioning versus ill-posedness for the boundary controllability of the heat equation

2015 ◽  
Vol 23 (4) ◽  
Author(s):  
Faker Ben Belgacem ◽  
Sidi Mahmoud Kaber
Keyword(s):  
Heat Equation ◽  
Exact Controllability ◽  
Adjoint Equations ◽  
Boundary Controllability ◽  
The Troubles ◽  
Hilbert Uniqueness Method ◽  
Ill Posed ◽  
Uniqueness Method ◽  
Ill Conditioning

AbstractIll-posedness and/or ill-conditioning are features users have to deal with appropriately in the controllability of diffusion problems for secure and reliable outputs. We investigate those issues in the case of a boundary Dirichlet control, in an attempt to underline the origin of the troubles arising in the numerical computations and to shed some light on the difficulties to obtain good quality simulations. The exact-controllability is severely ill-posed while, in spite of its well-posedness, the null-controllability turns out to be very badly ill-conditioned. Theoretical and numerical results are stated on the heat equation in one dimension to illustrate the specific instabilities of each problem. The main tools used here are first a characterization of the subspace where the HUM (Hilbert Uniqueness Method) control lies and the study of the spectrum of some structured matrices, of Pick and Löwner type, obtained from the Fourier calculations on the state and adjoint equations.

scholarly journals Exact Neumann boundary controllability for problems of transmission of the wave equation

1999 ◽  
Vol 41 (1) ◽  
pp. 125-139 ◽  
Author(s):  
WEIJIU LIU ◽  
GRAHAM H. WILLIAMS
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Exact Controllability ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Boundary Controllability ◽  
Hilbert Uniqueness Method ◽  
Uniqueness Method ◽  
Initial States

Using the Hilbert Uniqueness Method, we study the problem of exact controllability in Neumann boundary conditions for problems of transmission of the wave equation. We prove that this system is exactly controllable for all initial states in L2(Ω)×(H1(Ω))′.

scholarly journals A note on the exact boundary controllability for an imperfect transmission problem

2021 ◽  
Author(s):  
S. Monsurrò ◽  
A. K. Nandakumaran ◽  
C. Perugia
Keyword(s):  
Exact Controllability ◽  
Dirichlet Condition ◽  
Adjoint Problem ◽  
Imperfect Interface ◽  
External Boundary ◽  
Exact Boundary Controllability ◽  
Multipliers Method ◽  
Exact Boundary ◽  
Hilbert Uniqueness Method ◽  
Uniqueness Method

AbstractIn this note, we consider a hyperbolic system of equations in a domain made up of two components. We prescribe a homogeneous Dirichlet condition on the exterior boundary and a jump of the displacement proportional to the conormal derivatives on the interface. This last condition is the mathematical interpretation of an imperfect interface. We apply a control on the external boundary and, by means of the Hilbert Uniqueness Method, introduced by J. L. Lions, we study the related boundary exact controllability problem. The key point is to derive an observability inequality by using the so called Lagrange multipliers method, and then to construct the exact control through the solution of an adjoint problem. Eventually, we prove a lower bound for the control time which depends on the geometry of the domain, on the coefficients matrix and on the proportionality between the jump of the solution and the conormal derivatives on the interface.

scholarly journals Observability and uniqueness theorem for a coupled hyperbolic system

2000 ◽  
Vol 24 (6) ◽  
pp. 423-432 ◽  
Author(s):  
Boris V. Kapitonov ◽  
Joel S. Souza
Keyword(s):  
Uniqueness Theorem ◽  
Lagrange Multiplier ◽  
Hyperbolic System ◽  
Exact Controllability ◽  
Lagrange Multiplier Method ◽  
Multiplier Method ◽  
Inverse Inequality ◽  
Hilbert Uniqueness Method ◽  
Uniqueness Method

We deal with the inverse inequality for a coupled hyperbolic system with dissipation. The inverse inequality is an indispensable inequality that appears in the Hilbert Uniqueness Method (HUM), to establish equivalence of norms which guarantees uniqueness and boundary exact controllability results. The term observability is due to the mathematician Ho (1986) who used it in his works relating it to the inverse inequality. We obtain the inverse inequality by the Lagrange multiplier method under certain conditions.

scholarly journals Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects

2018 ◽  
Vol 36 (4) ◽  
pp. 1199-1235 ◽  
Author(s):  
Umberto Biccari ◽  
Víctor Hernández-Santamaría
Keyword(s):  
Heat Equation ◽  
Approximate Controllability ◽  
Fractional Laplacian ◽  
Controllability Problem ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
One Dimensional ◽  
Fractional Heat Equation ◽  
Hilbert Uniqueness Method ◽  
Uniqueness Method

Abstract We analyse the controllability problem for a one-dimensional heat equation involving the fractional Laplacian $(-d_x^{\,2})^{s}$ on the interval $(-1,1)$. Using classical results and techniques, we show that, acting from an open subset $\omega \subset (-1,1)$, the problem is null-controllable for $s>1/2$ and that for $s\leqslant 1/2$ we only have approximate controllability. Moreover, we deal with the numerical computation of the control employing the penalized Hilbert Uniqueness Method and a finite element scheme for the approximation of the solution to the corresponding elliptic equation. We present several experiments confirming the expected controllability properties.

scholarly journals Controllability for a Wave Equation with Moving Boundary

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Lizhi Cui ◽  
Libo Song
Keyword(s):  
Wave Equation ◽  
Dirichlet Boundary ◽  
Moving Boundary ◽  
Energy Estimates ◽  
Boundary Controllability ◽  
One Dimensional ◽  
Elastic String ◽  
Hilbert Uniqueness Method ◽  
Uniqueness Method ◽  
Dimensional Wave

We investigate the controllability for a one-dimensional wave equation in domains with moving boundary. This model characterizes small vibrations of a stretched elastic string when one of the two endpoints varies. When the speed of the moving endpoint is less than1-1/e, by Hilbert uniqueness method, sidewise energy estimates method, and multiplier method, we get partial Dirichlet boundary controllability. Moreover, we will give a sharper estimate on controllability time that only depends on the speed of the moving endpoint.

scholarly journals Machine learning powered ellipsometry

2021 ◽  
Vol 10 (1) ◽  
Author(s):  
Jinchao Liu ◽  
Di Zhang ◽  
Dianqiang Yu ◽  
Mengxin Ren ◽  
Jingjun Xu
Keyword(s):  
Machine Learning ◽  
Optical Constants ◽  
Optical Characterization ◽  
Superior Performance ◽  
Trial And Error ◽  
Powerful Method ◽  
Human In The Loop ◽  
Ill Posed ◽  
Fully Automatic

AbstractEllipsometry is a powerful method for determining both the optical constants and thickness of thin films. For decades, solutions to ill-posed inverse ellipsometric problems require substantial human–expert intervention and have become essentially human-in-the-loop trial-and-error processes that are not only tedious and time-consuming but also limit the applicability of ellipsometry. Here, we demonstrate a machine learning based approach for solving ellipsometric problems in an unambiguous and fully automatic manner while showing superior performance. The proposed approach is experimentally validated by using a broad range of films covering categories of metals, semiconductors, and dielectrics. This method is compatible with existing ellipsometers and paves the way for realizing the automatic, rapid, high-throughput optical characterization of films.

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