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Boundary controllability for a degenerate beam equation
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Abstract
The paper deals with the controllability of a degenerate beam equation. In particular, we assume that the left end of the beam is fixed, while a suitable control f acts on the right end of it. As a first step we prove the existence of a solution for the homogeneous problem, then we prove some estimates on its energy. Thanks to them we prove an observability inequality and, using the notion of solution by transposition, we prove that the initial problem is null controllable.
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In this article we consider the fourth-order operators and in divergence and non divergence form, where degenerates in an interior point of the interval. Using the semigroup technique, under suitable assumptions on a, we study the generation property of these operators associated to generalized Wentzell boundary conditions. We prove the well posedness of the corresponding parabolic problems.
This article concerns a nonlinear system modeling the bending vibrations
of a nonlinear beam of length . First, we derive the existence of
long time solutions near an equilibrium. Then we prove that the nonlinear
beam is locally exact controllable around the equilibrium in
and with control functions in . The approach we used are open
mapping theorem, local controllability established by linearization,
and the induction.
We study a wave equation in one space dimension with a general diffusion
coefficient which degenerates on part of the boundary. Degeneracy is measured
by a real parameter . We establish observability inequalities for
weakly (when ) as well as strongly (when )
degenerate equations. We also prove a negative result when the diffusion
coefficient degenerates too violently (i.e. when ) and the blow-up of
the observability time when converges to 2 from below. Thus, using
the HUM method we deduce the exact controllability of the corresponding
degenerate control problem when . We conclude the paper by
studying the boundary stabilization of the degenerate linearly damped wave
equation and show that a suitable boundary feedback stabilizes the system
exponentially. We extend this stability analysis to the degenerate nonlinearly
boundary damped wave equation, for an arbitrarily growing nonlinear feedback
close to the origin. This analysis proves that the degeneracy does not affect
the optimal energy decay rates at large time. We apply the optimal-weight
convexity method of \cite{alaamo2005, alajde2010} together with the results of
the previous section, to perform this stability analysis.
We consider a finite difference semi-discrete scheme for the approximation of the boundary controls of
a 1-D equation modelling the transversal vibrations of a hinged beam. It is known that, due to the high
frequency numerical spurious oscillations, the uniform (with respect to the mesh-size) controllability property
of the semi-discrete model fails in the natural setting. Consequently, the convergence of the approximate
controls corresponding to initial data in the finite energy space cannot be guaranteed. We prove that,
by adding a vanishing numerical viscosity, the uniform controllability property and the convergence of the
scheme is ensured.
The boundary controllability of a class of one-dimensional degenerate equations is studied in this paper. The novelty of this work is that the control acts through the part of the boundary where degeneracy occurs. First we consider a class of degenerate hyperbolic equations. Then, we prove sharp observability estimates for these equations using nonharmonic Fourier series. The transmutation method yields a result for the corresponding class of degenerate parabolic equations.
We give null controllability results for some degenerate parabolic equations in non divergence form on a bounded interval.
In particular, the coefficient of the second order term degenerates at the extreme points of the domain. For this reason,
we obtain an observability inequality for the adjoint problem. Then we prove Carleman estimates for such a problem. Finally,
in a standard way, we deduce null controllability also for semilinear equations.
We study the exact controllability of a nonlinear plate equation by the means of a control which acts on an internal region
of the plate. The main result asserts that this system is locally exactly controllable if the associated linear Euler–Bernoulli
system is exactly controllable. In particular, for rectangular domains, we obtain that the Berger system is locally exactly
controllable in arbitrarily small time and for every open and nonempty control region.
Exact controllability is studied for distributed systems, of hyperbolic type or for Petrowsky systems (like plate equations). The control is a boundary control or a local distributed control. Exact controllability consists in trying to drive the system to rest in a given finite time. The solution of the problems depends on the function spaces where the initial data are taken, and also depends on the function space where the control can be chosen. A systematic method (named HUM, for Hilbert Uniqueness Method) is introduced. It is based on uniqueness results (classical or new) and on Hilbert spaces constructed (in infinitely many ways) by using Uniqueness. A number of applications are indicated. Nonlinear Riccati type PDEs are obtained. Finally, we consider how all this behaves for perturbed systems.
This paper considers the Euler-Bernoulli problem with boundary controls g1, g2 in the Dirichlet and Neumann boundary conditions, respectively. Several exact controllability results are shown, including the following. The problem is exactly controllable in an arbitrarily short time T>0 in the space (of maximal regularity) H-1(Ω) × V′, V as specified, (i) with boundary controls g1 member of L2(Σ), g2≡0 under some geometrical conditions on Ω; (ii) with boundary controls g1 member of L2(Σ) and g2 member of L2(0, T; H-1(Γ)) without geometrical conditions on Ω. A direct approach is given, based on an operator model for the problem and on multiplier techniques.
A vibrating plate is here taken to satisfy the model equation:u
tt + 2u = 0 (where
2u:= (u); = Laplacian) with boundary conditions of the form:u
v = 0 and(u)
v = = control. Thus, the state is the pair [u, u
t] and controllability means existence of on := (0,T) transfering any[u, u
t]0 to any[u, u
t]T. The formulation is given by eigenfunction expansion and duality. The substantive results apply to a rectangular plate. For largeT one has such controllability with = O(T
–1/2). More surprising is that (based on a harmonic analysis estimate [11]) one has controllability for arbitrarily short times (in contrast to the wave equation:u
tt = u) with log = O(T
–1) asT0. Some related results on minimum time control are also included.
We propose a finite difference semi-discrete scheme for the approximation of the boundary exact controllability problem of the 1-D beam equation modelling the transversal vibrations of a beam with fixed ends. First of all we show that, due to the high frequency spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural functional setting. We then prove that there are two ways of restoring the uniform controllability property: a) filtering the high frequencies, i.e. controlling projections on subspaces where the high frequencies have been filtered; b) adding an extra boundary control to kill the spurious high frequency oscillations. In both cases the convergence of controls and controlled solutions is proved in weak and strong topologies, under suitable assumptions on the convergence of the initial data.
We consider the exact controllability of a hybrid
system consisting of an elastic beam, clamped at one end and attached
at the other end to a
rigid antenna. Such a system is governed by one partial
differential equation and two ordinary differential equations. Using the
HUM method, we prove that the hybrid system is exactly
controllable in an arbitrarily short time in the usual energy space.
Boundary controllability for a degenerate wave equation in non divergence form with drift
- I Boutaayamou
- G Fragnelli
- D Mugnai
I. Boutaayamou, G. Fragnelli, D. Mugnai, Boundary controllability for a
degenerate wave equation in non divergence form with drift, to appear in
SIAM J. Control Optim.
A degenerate operator in non divergence form
- A Camasta
- G Fragnelli
A. Camasta, G. Fragnelli, A degenerate operator in non divergence form,
Recent Advances in Mathematical Analysis, Trends in Mathematics,
https://doi.org/10.1007/978-3-031-20021-2.
Degenerate fourth order parabolic equations with Neumann boundary conditions
- A Camasta
- G Fragnelli
A. Camasta, G. Fragnelli, Degenerate fourth order parabolic equations with Neumann boundary conditions, submitted for publication,
arXiv:2203.02739.
Global smooth solutions for a nonlinear beam with boundary input and output
- P F Yao
- G Weiss
P.F. Yao, G. Weiss, Global smooth solutions for a nonlinear beam with
boundary input and output, SIAM Journal on Control and Optimization
45 (2007), 1931-1964.