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.

Exact controllability for a model of a multidimensional flexible structure

1993 ◽  
Vol 123 (2) ◽  
pp. 323-344 ◽  
Author(s):  
Jean Pierre Puel ◽  
Enrique Zuazua
Keyword(s):  
Wave Equation ◽  
Simple Model ◽  
Exact Controllability ◽  
Flexible Structure ◽  
Junction Region ◽  
Exact Boundary Controllability ◽  
Boundary Controllability ◽  
One Dimensional ◽  
Exact Boundary ◽  
Simple Boundary

SynopsisA simple model of a vibrating multidimensional structure made of a n-dimensional body and a one-dimensional straight string is introduced. In both regions (n-dimensional body and a onedimensional string) the state is assumed to satisfy the wave equation. Simple boundary conditions are introduced at the junction. These conditions, in the absence of control, ensure conservation of the total energy of the system and imply some rigidity of the boundary of the n-d body on a neighbourhood of the junction. The exact boundary controllability of the system is proved by means of a Dirichlet control supported on a subset of the boundary of the n-d domain which excludes the junction region. Some extensions are discussed at the end of the paper.

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 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))′.

EXACT BOUNDARY CONTROLLABILITY OF A NONLINEAR SHALLOW SPHERICAL SHELL

1998 ◽  
Vol 08 (06) ◽  
pp. 927-955
Author(s):  
MARY E. BRADLEY ◽  
IRENA LASIECKA
Keyword(s):  
Nonlinear System ◽  
Spherical Shell ◽  
Exact Controllability ◽  
Finite Energy ◽  
Shallow Spherical Shell ◽  
Geometric Condition ◽  
Shear Forces ◽  
Exact Boundary Controllability ◽  
Boundary Controllability ◽  
Exact Boundary

We consider the problem of boundary exact controllability of a coupled nonlinear system which describes vibrations of a thin, shallow, spherical shell. We show that under the geometric condition of "shallowness", which restricts the curvature with respect to the thickness, the system is exactly controllable in the natural "finite energy" space by means of L2 controls. This controllability is produced via moments and shear forces applied to the edge of the shell.

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 On the Exact Boundary Control for the Linear Klein-Gordon Equation in Non-cylindrical Domains

2020 ◽  
Vol 21 (2) ◽  
pp. 371
Author(s):  
R. S. O. Nunes
Keyword(s):  
Initial Data ◽  
Wave Operator ◽  
Gordon Equation ◽  
Controllability Problem ◽  
Exact Boundary Controllability ◽  
Boundary Controllability ◽  
Exact Boundary ◽  
Cylindrical Domains ◽  
Klein Gordon ◽  
Klein Gordon Equation

The purpose of this paper is to study an exact boundary controllability problem in noncylindrical domains for the linear Klein-Gordon equation. Here, we work near of the extension techniques presented By J. Lagnese in [12] which is based in the Russell’s controllability method. The control time is obtained in any time greater then the value of the diameter of the domain on which the initial data are supported. The control is square integrable and acts on whole boundary and it is given by conormal derivative associated with the above-referenced wave operator.

Sign in / Sign up

Export Citation Format

Share Document