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Evolution Equations & Control Theory

2015 , Volume 4 , Issue 1

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Boundary feedback stabilization of a chain of serially connected strings
Kaïs Ammari and Denis Mercier
2015, 4(1): 1-19 doi: 10.3934/eect.2015.4.1 +[Abstract](189) +[PDF](489.7KB)
Abstract:
We consider $N$ strings connected one to another and forming a particular network which is a chain of strings. We study a stabilization problem and precisely we prove that the energy of the solutions of the dissipative system decays exponentially to zero when the time tends to infinity, independently of the densities of the strings. Our technique is based on a frequency domain method and a special analysis for the resolvent. Moreover, by the same approach, we study the transfer function associated to the chain of strings and the stability of the Schrödinger system.
Optimal energy decay rate of Rayleigh beam equation with only one boundary control force
Maya Bassam, Denis Mercier and Ali Wehbe
2015, 4(1): 21-38 doi: 10.3934/eect.2015.4.21 +[Abstract](253) +[PDF](475.0KB)
Abstract:
We consider a clamped Rayleigh beam equation subject to only one boundary control force. Using an explicit approximation, we first give the asymptotic expansion of eigenvalues and eigenfunctions of the undamped underlying system. We next establish a polynomial energy decay rate for smooth initial data via an observability inequality of the corresponding undamped problem combined with a boundedness property of the transfer function of the associated undamped problem. Finally, by a frequency domain approach, using the real part of the asymptotic expansion of eigenvalues of the infinitesimal generator of the associated semigroup, we prove that the obtained energy decay rate is optimal.
Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation
Samira Boussaïd, Danielle Hilhorst and Thanh Nam Nguyen
2015, 4(1): 39-59 doi: 10.3934/eect.2015.4.39 +[Abstract](315) +[PDF](477.7KB)
Abstract:
We consider a nonlocal reaction-diffusion equation with mass conservation, which was originally proposed by Rubinstein and Sternberg as a model for phase separation in a binary mixture. We study the large time behavior of the solution and show that it converges to a stationary solution as $t$ tends to infinity. We also evaluate the rate of convergence. In some special case, we show that the limit solution is constant.
Local boundedness property for parabolic BVP's and the Gaussian upper bound for their Green functions
Mourad Choulli
2015, 4(1): 61-67 doi: 10.3934/eect.2015.4.61 +[Abstract](191) +[PDF](307.4KB)
Abstract:
In the present note, we give a concise proof for the equivalence between the local boundedness property for parabolic Dirichlet BVP's and the gaussian upper bound for their Green functions. The parabolic equations we consider are of general divergence form and our proof is essentially based on the gaussian upper bound by Daners [2] and a Caccioppoli's type inequality. We also show how the same analysis enables us to get a weaker version of the local boundedness property for parabolic Neumann BVP's assuming that the corresponding Green functions satisfy a gaussian upper bound.
The $L^p$-approach to the fluid-rigid body interaction problem for compressible fluids
Matthias Hieber and Miho Murata
2015, 4(1): 69-87 doi: 10.3934/eect.2015.4.69 +[Abstract](242) +[PDF](431.3KB)
Abstract:
Consider the system of equations describing the motion of a rigid body immersed in a viscous, compressible fluid within the barotropic regime. It is shown that this system admits a unique, local strong solution within the $L^p$-setting.
Global stabilization of the Navier-Stokes equations around an unstable equilibrium state with a boundary feedback controller
Evrad M. D. Ngom, Abdou Sène and Daniel Y. Le Roux
2015, 4(1): 89-106 doi: 10.3934/eect.2015.4.89 +[Abstract](277) +[PDF](428.1KB)
Abstract:
This paper presents a global stabilization for the two and three-dimensional Navier-Stokes equations in a bounded domain $\Omega$ around a given unstable equilibrium state, by means of a boundary normal feedback control. The control is expressed in terms of the velocity field by using a non-linear feedback law. In order to determine the feedback control law, we consider an extended system coupling the equations governing the perturbation with an equation satisfied by the control on the domain boundary. By using the Faedo-Galerkin method and a priori estimation techniques, a stabilizing boundary control is built. This control law ensures a decrease of the energy of the controlled discrete system. A compactness result then allows us to pass to the limit in the system satisfied by the approximated solutions.
Backward uniqueness for linearized compressible flow
Michael Renardy
2015, 4(1): 107-113 doi: 10.3934/eect.2015.4.107 +[Abstract](212) +[PDF](299.1KB)
Abstract:
We prove that a $C_0$-semigroup of operators $\exp(At)$ satisfies backward uniqueness if the resolvent of $A$ exists on a ray $z=re^{i\theta}$ in the left half plane ($\pi/2<\theta\le \pi$) and satisfies a bound $\|(A-z I)^{-1}\|\le C\exp(|z|^\alpha)$, $\alpha<1$ on this ray. The proof of this result is based on the Phragmen-Lindelöf theorem. The result is applied to the linearized compressible Navier-Stokes equations in one space dimension and to the wave equation with linear damping and absorbing boundary condition.

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