Convergence Rates of the Hilbert Uniqueness Method via Tikhonov Regularization

1999 ◽  
Vol 103 (3) ◽  
pp. 657-673 ◽  
Author(s):  
S. Kindermann
Keyword(s):  
Tikhonov Regularization ◽  
Convergence Rates ◽  
Hilbert Uniqueness Method ◽  
Uniqueness Method

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 An Output Controllability Problem for Semilinear Distributed Hyperbolic Systems

2007 ◽  
Vol 17 (4) ◽  
pp. 437-448 ◽  
Author(s):  
E Zerrik ◽  
R Larhrissi ◽  
H Bourray
Keyword(s):  
Fixed Point ◽  
Generalized Inverse ◽  
Hyperbolic Systems ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Controllability Problem ◽  
Hilbert Uniqueness Method ◽  
Uniqueness Method ◽  
Output Controllability ◽  
Regional Controllability

An Output Controllability Problem for Semilinear Distributed Hyperbolic SystemsThe paper aims at extending the notion of regional controllability developed for linear systems to the semilinear hyperbolic case. We begin with an asymptotically linear system and the approach is based on an extension of the Hilbert uniqueness method and Schauder's fixed point theorem. The analytical case is then tackled using generalized inverse techniques and converted to a fixed point problem leading to an algorithm which is successfully implemented numerically and illustrated with examples.

Convergence rates for Tikhonov regularization based on range inclusions

2005 ◽  
Vol 21 (3) ◽  
pp. 805-820 ◽  
Author(s):  
Bernd Hofmann ◽  
Masahiro Yamamoto
Keyword(s):  
Tikhonov Regularization ◽  
Convergence Rates

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.

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