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Evolution Equations & Control Theory

2013 , Volume 2 , Issue 4

Special issue dedicated to Walter Littman on the occasion of his retirement

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Preface
Stephen W. Taylor and Roberto Triggiani
2013, 2(4): i-ii doi: 10.3934/eect.2013.2.4i +[Abstract](245) +[PDF](305.1KB)
Abstract:
This volume is dedicated to Walter Littman on the occasion of his retirement.

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A problem of boundary controllability for a plate
Orazio Arena
2013, 2(4): 557-562 doi: 10.3934/eect.2013.2.557 +[Abstract](241) +[PDF](309.5KB)
Abstract:
The boundary controllability problem, here discussed, might be described by a two-dimensional space equation modeling, at the same time $t$, different physical phenomena in a composite solid made of different materials. These phenomena may be governed, at the same time $t$, for example, by the heat equation and by the Schrödinger equation in two separate regions. Interface transmission conditions are imposed.
Fluid-structure interaction with and without internal dissipation of the structure: A contrast study in stability
George Avalos and Roberto Triggiani
2013, 2(4): 563-598 doi: 10.3934/eect.2013.2.563 +[Abstract](446) +[PDF](827.5KB)
Abstract:
We consider a coupled parabolic--hyperbolic PDE system arising in fluid--structure interaction, where the coupling is exercised at the interface between the two media. This paper is a study in contrast on stability properties of the overall coupled system under two scenarios: the case with interior dissipation of the structure, and the case without. In the first case, uniform stabilization is achieved (by a $\lambda$-domain analysis) without geometrical conditions on the structure, but only on an explicitly identified space $Ĥ$ of codimension one with respect to the original energy state space $H$ where semigroup well-posedness holds. In the second case, only rational (a fortiori strong) stability is possible, again only on the space $Ĥ$, however, under geometrical conditions of the structure, which e.g., exclude a sphere. Many classes of good geometries are identified. Recent papers [6,9] show uniform stabilization on all of $H$, and without geometrical conditions; however, with dissipation at the boundary interface.
On Landau-Lifshitz equations of no-exchange energy models in ferromagnetics
Wei Deng and Baisheng Yan
2013, 2(4): 599-620 doi: 10.3934/eect.2013.2.599 +[Abstract](418) +[PDF](479.7KB)
Abstract:
In this paper, we study Landau-Lifshitz equations of ferromagnetism with a total energy that does not include a so-called exchange energy. Many problems, including existence, stability, regularity and asymptotic behaviors, have been extensively studied for such equations of models with the exchange energy. The problems turn out quite different and challenging for Landau-Lifshitz equations of no-exchange energy models because the usual methods based on certain compactness do not apply. We present a new method for the existence of global weak solution to the Landau-Lifshitz equation of no-exchange energy models based on the existence of regular solutions for smooth data and certain stability of the solutions. We also study higher time regularity, energy identity and asymptotic behaviors in some special cases for weak solutions.
A remark on Littman's method of boundary controllability
Matthias Eller
2013, 2(4): 621-630 doi: 10.3934/eect.2013.2.621 +[Abstract](295) +[PDF](337.3KB)
Abstract:
We extend the method of exact boundary controllability of strictly hyperbolic equations developed by W. Littman [22,23] to a large class of hyperbolic systems with constant coefficients. Our approach is based on the knowledge of the singularities of the fundamental solution of hyperbolic operators.
Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions
Nicolas Fourrier and Irena Lasiecka
2013, 2(4): 631-667 doi: 10.3934/eect.2013.2.631 +[Abstract](632) +[PDF](803.7KB)
Abstract:
We present an analysis of regularity and stability of solutions corresponding to wave equation with dynamic boundary conditions. It has been known since the pioneering work by [26, 27, 30] that addition of dynamics to the boundary may change drastically both regularity and stability properties of the underlying system. We shall investigate these properties in the context of wave equation with the damping affecting either the interior dynamics or the boundary dynamics or both.
    This leads to a consideration of a wave equation acting on a bounded 3-d domain coupled with another second order dynamics acting on the boundary. The wave equation is equipped with a viscoelastic damping, zero Dirichlet boundary conditions on a portion of the boundary and dynamic boundary conditions. These are general Wentzell type of boundary conditions which describe wave equation oscillating on a tangent manifold of a lower dimension. We shall examine regularity and stability properties of the resulting system -as a function of strength and location of the dissipation. Properties such as well-posedness of finite energy solutions, analyticity of the associated semigroup, strong and uniform stability will be discussed.
    The results obtained analytically are illustrated by numerical analysis. The latter shows the impact of various types of dissipation on the spectrum of the generator as well as the dynamic behavior of the solution on a rectangular domain.
Control of blow-up singularities for nonlinear wave equations
Satyanad Kichenassamy
2013, 2(4): 669-677 doi: 10.3934/eect.2013.2.669 +[Abstract](339) +[PDF](373.0KB)
Abstract:
While the global boundary control of nonlinear wave equations that exhibit blow-up is generally impossible, we show on a typical example, motivated by laser breakdown, that it is possible to control solutions with small data so that they blow up on a prescribed compact set bounded away from the boundary of the domain. This is achieved using the representation of singular solutions with prescribed blow-up surface given by Fuchsian reduction. We outline on this example simple methods that may be of wider applicability.
Uniqueness and Hölder type stability of continuation for the linear thermoelasticity system with residual stress
Nanhee Kim
2013, 2(4): 679-693 doi: 10.3934/eect.2013.2.679 +[Abstract](260) +[PDF](370.5KB)
Abstract:
By introducing some auxiliary functions, an elasticity system with thermal effects becomes a coupled hyperbolic-parabolic system. Using this reduced system, we obtain a Carleman estimate with two large parameters for the linear thermoelasticity system with residual stress which is the basic tool for showing stability estimates in the lateral Cauchy problem.
Uniform stabilization of a multilayer Rao-Nakra sandwich beam
A. Özkan Özer and Scott W. Hansen
2013, 2(4): 695-710 doi: 10.3934/eect.2013.2.695 +[Abstract](278) +[PDF](450.6KB)
Abstract:
We consider the problem of boundary feedback stabilization of a multilayer Rao-Nakra sandwich beam. We show that the eigenfunctions of the decoupled system form a Riesz basis. This allows us to deduce that the decoupled system is exponentially stable. Since the coupling terms are compact, the exponential stability of the coupled system follows from the strong stability of the coupled system, which is proved using a unique continuation result for the overdetermined homogenous system in the case of zero feedback.
Trace properties of certain damped linear elastic systems
David L. Russell
2013, 2(4): 711-721 doi: 10.3934/eect.2013.2.711 +[Abstract](348) +[PDF](320.7KB)
Abstract:
We study the spectrum of a damped linear elastic system with discrete eigenvalues, showing the relationship between the sum of the real parts of the eigenvalues of the (generally unbounded) generator and the trace of the damping operator, assuming the latter to be a trace type operator. Some relationships between the sequence of eigenvectors and a corresponding orthonormal sequence, constructed by means of a variant of the Gram-Schmidt method, are also explored. A simple hybrid system is presented as an example of application.
Optimal control of a diffusion/reaction/switching system
Thomas I. Seidman
2013, 2(4): 723-731 doi: 10.3934/eect.2013.2.723 +[Abstract](283) +[PDF](328.5KB)
Abstract:
We consider an optimal control problem involving the use of bacteria for pollution removal where the model assumes the bacteria switch instantaneously between active and dormant states, determined by threshold sensitivity to the local concentration $v$ of a diffusing critical nutrient; compare [7], [3], [6] in which nutrient transport is convective. It is shown that the direct problem has a solution for each boundary control $ψ = ∂v/∂n$ and that optimal controls exist, minimizing a combination of residual pollutant and aggregated cost of the nutrient.
Locally smooth unitary groups and applications to boundary control of PDEs
Stephen W. Taylor
2013, 2(4): 733-740 doi: 10.3934/eect.2013.2.733 +[Abstract](278) +[PDF](344.5KB)
Abstract:
Let $\mathcal{P}$ be the projection operator for a closed subspace $\mathcal{S}$ of a Hilbert space $\mathcal{H}$ and let $U$ be a unitary operator on $\mathcal{H}$. We consider the questions
    1. Under what conditions is $\mathcal{P}U\mathcal{P}$ a strict contraction?
    2. If $g$, $h\in \mathcal{S}$, can we find $f\in \mathcal{H}$ such that $\mathcal{P}f=g$ and $\mathcal{P}Uf=h$?
The results are abstract versions and generalisations of results developed for boundary control of partial differential equations. We discuss how these results can be used as tools in the direct construction of boundary controls.
The approximate controllability of a model for mutant selection
Hans Weinberger
2013, 2(4): 741-747 doi: 10.3934/eect.2013.2.741 +[Abstract](261) +[PDF](275.2KB)
Abstract:
It is shown that the problem of eliminating a less-fit allele by allowing a mixture of genotypes whose densities satisfy a system of reaction-diffusion equations with population control to evolve in a reactor with impenetrable walls is approximately controllable.
Study on the order of events in optimal control of a harvesting problem modeled by integrodifference equations
Peng Zhong and Suzanne Lenhart
2013, 2(4): 749-769 doi: 10.3934/eect.2013.2.749 +[Abstract](256) +[PDF](491.8KB)
Abstract:
Integrodifference equations are discrete in time and continuous in space, and are used to model populations that are growing at discrete times, and dispersing spatially. A harvesting problem modeled by integrodifference equations involves three events: growth, dispersal and harvesting. The order of arranging the three events affects the optimized harvesting behavior. In this paper we investigate all six possible cases of orders of events, study the equivalences among them under certain conditions, and show how the six cases can be reduced to three cases.

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