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Evolution Equations & Control Theory

2013 , Volume 2 , Issue 1

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Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems
Farah Abdallah, Denis Mercier and Serge Nicaise
2013, 2(1): 1-33 doi: 10.3934/eect.2013.2.1 +[Abstract](64) +[PDF](756.6KB)
Abstract:
In this paper, we consider two damped wave problems for which the damping terms are allowed to change their sign. Using a careful spectral analysis, we find critical values of the damping coefficients for which the problem becomes exponentially or polynomially stable up to these critical values.
Vibrations of a damped extensible beam between two stops
Alessia Berti and Maria Grazia Naso
2013, 2(1): 35-54 doi: 10.3934/eect.2013.2.35 +[Abstract](87) +[PDF](404.3KB)
Abstract:
A PDE system modeling the dynamics of an extensible beam having one of its ends constrained between two stops is considered. The existence of a weak global-in-time solution is established by a penalization method. In addition, the asymptotic behavior of such a solution is analyzed and the exponential decay rate for the related energy is shown.
Sensitivity analysis for a free boundary fluid-elasticity interaction
Lorena Bociu and Jean-Paul Zolésio
2013, 2(1): 55-79 doi: 10.3934/eect.2013.2.55 +[Abstract](80) +[PDF](452.3KB)
Abstract:
In this paper a total linearization is derived for the free boundary nonlinear elasticity - incompressible fluid interaction. The equations and the free boundary are linearized together and the new linearization turns out to be different from the usual coupling of classical linear models. New extra terms are present on the common interface, some of them involving the boundary curvatures. These terms play an important role in the final linearized system and can not be neglected.
Orbitally stable standing waves for the asymptotically linear one-dimensional NLS
François Genoud
2013, 2(1): 81-100 doi: 10.3934/eect.2013.2.81 +[Abstract](85) +[PDF](493.7KB)
Abstract:
In this article we study the one-dimensional, asymptotically linear, non-linear Schrödinger equation (NLS). We show the existence of a global smooth curve of standing waves for this problem, and we prove that these standing waves are orbitally stable. As far as we know, this is the first rigorous stability result for the asymptotically linear NLS. We also discuss an application of our results to self-focusing waveguides with a saturable refractive index.
Analyticity and regularity for a class of second order evolution equations
Alain Haraux and Mitsuharu Ôtani
2013, 2(1): 101-117 doi: 10.3934/eect.2013.2.101 +[Abstract](74) +[PDF](413.8KB)
Abstract:
The regularity conservation as well as the smoothing effect are studied for the equation $ u''+ Au+ cA^\alpha u' = 0$, where $A$ is a positive selfadjoint operator on a real Hilbert space $H$ and $\alpha\in (0, 1]; \,\, c >0$. When $\alpha\ge {1\over 2}$ the equation generates an analytic semigroup on $D(A^{1/2})\times H $ , and if $\alpha\in (0, {1\over 2})$ a weaker optimal smoothing property is established. Some conservation properties in other norms are also established, as a typical example, the strongly dissipative wave equation $u_{tt} - \Delta u -c\Delta u_t = 0$ with Dirichlet boundary conditions in a bounded domain is given, for which the space $C_0(\Omega)\times C_0(\Omega)$ is conserved for $t>0$, which presents a sharp contrast with the conservative case $u_{tt} - \Delta u = 0$ for which $C_0(\Omega)$-regularity can be lost even starting from an initial state $(u_0, 0)$ with $u_0\in C_0(\Omega)\cap C^1(\overline {\Omega})$.
A variational approach to approximate controls for system with essential spectrum: Application to membranal arch
Arnaud Münch
2013, 2(1): 119-151 doi: 10.3934/eect.2013.2.119 +[Abstract](67) +[PDF](5292.1KB)
Abstract:
We address the numerical approximation of boundary controls for systems of the form $\boldsymbol{y^{\prime\prime}}+\boldsymbol{A_M}\boldsymbol{y}=\boldsymbol{0}$ which models dynamical elastic shell structure. The membranal operator $\boldsymbol{A_M}$ is self-adjoint and of mixed order, so that it possesses a non empty and bounded essential spectrum $\sigma_{ess}(\boldsymbol{A_M})$. Consequently, the exact controllability does not hold uniformly with respect to the initial data. Thus the numerical computation of controls by the way of dual approach and gradient method may fail, even if the initial data belongs to the orthogonal of the space spanned by the eigenfunctions associated with $\sigma_{ess}(\boldsymbol{A_M})$. In this work, we adapt a variational approach introduced in [Pablo Pedregal, Inverse Problems (26) 015004 (2010)] for the wave equation and obtain a robust method of approximation. This new approach does not require any information on the spectrum of the operator $\boldsymbol{A_M}$. We also show that it allows to extract, from any initial data $(\boldsymbol{y^0},\boldsymbol{y^1})$, a controllable component for the mixed order system. We illustrate these properties with some numerical experiments. We also consider a relaxed controllability case for which uniform property holds.
Simultaneous stabilization of a system of interacting plate and membrane
Louis Tebou
2013, 2(1): 153-172 doi: 10.3934/eect.2013.2.153 +[Abstract](59) +[PDF](149.5KB)
Abstract:
We consider a system consisting of a wave equation and a plate equation, where the rotational forces may be accounted for, in a bounded domain. The system is coupled through the dissipation which is localized in an appropriate portion of the domain under consideration. First, we show that this system is strongly stable; the feedback control region is arbitrarily small when the rotational inertia is greater than or equal to one, but it is required that its boundary intersect a nonnegligible portion of the boundary of the domain under consideration if the rotational inertia is less than one. Next we show that the system accounting for rotational forces is not exponentially stable. Afterwards, using a constructive frequency domain method, we show that the system with no rotational forces is exponentially stable provided that the feedback control region is large enough. New uniqueness and controllability results are derived.
Approximation of a semigroup model of anomalous diffusion in a bounded set
Stephen Thompson and Thomas I. Seidman
2013, 2(1): 173-192 doi: 10.3934/eect.2013.2.173 +[Abstract](95) +[PDF](429.1KB)
Abstract:
The convergence is established for a sequence of operator semigroups, where the limiting object is the transition semigroup for a reflected stable processes. For semilinear equations involving the generators of these transition semigroups, an approximation method is developed as well. This makes it possible to derive an a priori bound for solutions to these equations, and therefore prove global existence of solutions. An application to epidemiology is also given.

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