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Evolution Equations & Control Theory

2013 , Volume 2 , Issue 3

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Carleman Estimates and null controllability of coupled degenerate systems
El Mustapha Ait Ben Hassi, Farid Ammar khodja, Abdelkarim Hajjaj and Lahcen Maniar
2013, 2(3): 441-459 doi: 10.3934/eect.2013.2.441 +[Abstract](519) +[PDF](425.6KB)
In this paper, we study the null controllability of weakly degenerate parabolic systems with two different diffusion coefficients and one control force. To obtain this aim, we had to develop new global Carleman estimates for a degenerate parabolic equation, with weight functions different from the ones of [2], [10] and [31].
Asymptotics for a second order differential equation with a linear, slowly time-decaying damping term
Alain Haraux and Mohamed Ali Jendoubi
2013, 2(3): 461-470 doi: 10.3934/eect.2013.2.461 +[Abstract](417) +[PDF](333.5KB)
A gradient-like property is established for second order semilinear conservative systems in presence of a linear damping term which is asymptotically weak for large times. The result is obtained under the condition that the only critical points of the potential are absolute minima. The damping term may vanish on large intervals for arbitrarily large times and tends to $0$ at infinity, but not too fast (in a non-integrable way). When the potential satisfies an adapted, uniform, Łojasiewicz gradient inequality, convergence to equilibrium of all bounded solutions is shown, with examples in both analytic and non-analytic cases.
Traction, deformation and velocity of deformation in a viscoelastic string
Luciano Pandolfi
2013, 2(3): 471-493 doi: 10.3934/eect.2013.2.471 +[Abstract](385) +[PDF](420.8KB)
In this paper we consider a viscoelastic string whose deformation is controlled at one end. We study the relations and the controllability of the couples traction/velocity and traction/deformation and we show that the first couple behaves very like as in the purely elastic case, while new phenomena appears when studying the couple of the traction and the deformation. Namely, while traction and velocity are independent (for large time), traction and deformation are related at each time but the relation is not so strict. In fact we prove that an arbitrary number of ``Fourier'' components of the traction and, independently, of the deformation can be assigned at any time.
Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations
Pavel I. Plotnikov and Jan Sokolowski
2013, 2(3): 495-516 doi: 10.3934/eect.2013.2.495 +[Abstract](401) +[PDF](422.4KB)
The flow around a rigid obstacle is governed by the compressible Navier-Stokes equations. The nonhomogeneous Dirichlet problem is considered in a bounded domain in two spatial dimensions with a compact obstacle in its interior. The flight of the airflow is characterized by the work shape functional, to be minimized over a family of admissible obstacles. The lift of the airfoil is a given function of temporal variable and should be maintain closed to the flight scenario. The continuity of the work functional with respect to the shape of obstacle in two spatial dimensions is shown for a wide class of admissible obstacles compact with respect to the Kuratowski-Mosco convergence.
    The dependence of small perturbations of approximate solutions to the governing equations with respect to the boundary variations of obstacles is analyzed for the nonstationary state equation.
On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions
Cristina Pocci
2013, 2(3): 517-530 doi: 10.3934/eect.2013.2.517 +[Abstract](370) +[PDF](363.3KB)
In this paper we consider a geometric motion associated with the minimization of a functional which is the sum of a kinetic part of $p$-Laplacian type, a double well potential $\psi$ and a curvature term. In the case $p=2$, such a functional arises in connection with the image segmentation problem in computer vision theory. By means of matched asymptotic expansions, we show that the geometric motion can be approximated by the evolution of the zero level set of the solution of a nonlinear $p$-order equation. The singular limit depends on a complex way on the mean and Gaussian curvatures and the surface Laplacian of the mean curvature of the evolving front.
Energy methods for Hartree type equations with inverse-square potentials
Toshiyuki Suzuki
2013, 2(3): 531-542 doi: 10.3934/eect.2013.2.531 +[Abstract](421) +[PDF](199.6KB)
Nonlinear Schrödinger equations with nonlocal nonlinearities described by integral operators are considered. This generalizes usual Hartree type equations (HE)$_{0}$. We construct weak solutions to (HE)$_{a}$, $a\neq 0$, even if the kernel is of non-convolution type. The advantage of our methods is the applicability to the problem with strongly singular potential $a|x|^{-2}$ as a term in the linear part and with critical nonlinearity.
On the structural properties of an efficient feedback law
Ambroise Vest
2013, 2(3): 543-556 doi: 10.3934/eect.2013.2.543 +[Abstract](398) +[PDF](392.9KB)
We investigate some structural properties of an efficient feedback law that stabilize linear time-reversible systems with an arbitrarily large decay rate. After giving a short proof of the generation of a group by the closed-loop operator, we focus on the domain of the infinitesimal generator in order to illustrate the difference between a distributed control and a boundary control, the latter being technically more complex. We also give a new proof of the exponential decay of the solutions and we provide an explanation of the higher decay rate observed in some experiments.

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