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Evolution Equations & Control Theory

2014 , Volume 3 , Issue 2

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Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems
Björn Augner and  Birgit Jacob
2014, 3(2): 207-229 doi: 10.3934/eect.2014.3.207 +[Abstract](45) +[PDF](458.9KB)
Stability and stabilization of linear port-Hamiltonian systems on infinite-dimensional spaces are investigated. This class is general enough to include models of beams and waves as well as transport and Schrödinger equations with boundary control and observation. The analysis is based on the frequency domain method which gives new results for second order port-Hamiltonian systems and hybrid systems. Stabilizing SIP or SOP controllers are designed. The obtained results are applied to the Euler-Bernoulli beam.
On controllability of a linear elastic beam with memory under longitudinal load
Sergei A. Avdonin and  Boris P. Belinskiy
2014, 3(2): 231-245 doi: 10.3934/eect.2014.3.231 +[Abstract](49) +[PDF](421.5KB)
This work is motivated by the control problem for a linear elastic beam under a longitudinal load when the material of the beam has memory. We reduce the problem of controllability to a nonstandard moment problem. The solution of the latter problem is based on the Riesz basis property for a family of functions quadratically close to the nonharmonic exponentials. This result requires the detailed analysis of an integro--differential equation, and is of interest in itself for Function Theory.
Non-smooth unobservable states in control problem for the wave equation in $\mathbb{R}^3$
Mikhail I. Belishev and  Aleksei F. Vakulenko
2014, 3(2): 247-256 doi: 10.3934/eect.2014.3.247 +[Abstract](35) +[PDF](390.4KB)
The paper deals with a dynamical system \begin{align*} &u_{tt}-\Delta u=0, \qquad (x,t) \in {\mathbb R}^3 \times (-\infty,0) \\ &u \mid_{|x|<-t} =0 , \qquad t<0\\ &\lim_{s \to \infty} su((s+\tau)\omega,-s)=f(\tau,\omega), \qquad (\tau,\omega) \in [0,\infty)\times S^2\,, \end{align*} where $u=u^f(x,t)$ is a solution ( wave), $f \in {\mathcal F}:=L_2\left([0,\infty);L_2\left(S^2\right)\right)$ is a control. For the reachable sets ${\mathcal U}^\xi:=\{u^f(\cdot,-\xi)\,|\,\, f \in {\mathcal F}\}\,\,(\xi\geq 0)$, the embedding ${\mathcal U}^\xi \subset {\mathcal H}^\xi:=\{y \in L_2({\mathbb R}^3)\,|\,\,\,y|_{|x|<\xi}=0\}$ holds, whereas the subspaces ${\mathcal D}^\xi:={\mathcal H}^\xi \ominus {\mathcal U}^\xi$ of unreachable ( unobservable) states are nonzero for $\xi> 0$. There was a conjecture motivated by some geometrical optics arguments that the elements of ${\mathcal D}^\xi$ are $C^\infty$-smooth with respect to $|x|$. We provide rather unexpected counterexamples of $h\in {\mathcal D}^\xi$ with ${\rm sing\,supp\,}h \subset \{x\in{\mathbb R}^3|\,\,|x|=\xi_0>\xi\}$.
A vanishing diffusion limit in a nonstandard system of phase field equations
Pierluigi Colli , Gianni Gilardi , Pavel Krejčí and  Jürgen Sprekels
2014, 3(2): 257-275 doi: 10.3934/eect.2014.3.257 +[Abstract](37) +[PDF](289.0KB)
We are concerned with a nonstandard phase field model of Cahn-Hilliard type. The model, which was introduced by Podio-Guidugli (Ric. Mat. 2006), describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been recently investigated by Colli, Gilardi, Podio-Guidugli, and Sprekels in a series of papers: see, in particular, SIAM J. Appl. Math. 2011 and Boll. Unione Mat. Ital. 2012. In the latter contribution, the authors can treat the very general case in which the diffusivity coefficient of the parabolic PDE is allowed to depend nonlinearly on both variables. In the same framework, this paper investigates the asymptotic limit of the solutions to the initial-boundary value problems as the diffusion coefficient $\sigma$ in the equation governing the evolution of the order parameter tends to zero. We prove that such a limit actually exists and solves the limit problem, which couples a nonlinear PDE of parabolic type with an ODE accounting for the phase dynamics. In the case of a constant diffusivity, we are able to show uniqueness and to improve the regularity of the solution.
An integration model for two different ethnic groups
Mauro Fabrizio and  Jaime Munõz Rivera
2014, 3(2): 277-286 doi: 10.3934/eect.2014.3.277 +[Abstract](26) +[PDF](651.3KB)
For the purpose of studying the integration of two different ethnic populations, we compare their evolution with that of a mixture of two fluids. For this model we consider the concentration of only one species, whose evolution will be described by a Cahn-Hilliard equation. Instead, the separation between the two phases will be controlled by the educational levels of two components. Finally, we assume that the homogenization phase occurs when the mean of the cultural levels is greater then a critical value.
Observability of rectangular membranes and plates on small sets
Vilmos Komornik and  Paola Loreti
2014, 3(2): 287-304 doi: 10.3934/eect.2014.3.287 +[Abstract](29) +[PDF](532.3KB)
Since the works of Haraux and Jaffard we know that rectangular plates may be observed by subregions not satisfying the geometrical control condition. We improve these results by observing only on an arbitrarily short segment inside the domain. The estimates may be strengthened by observing on several well-chosen segments.
    In the second part of the paper we establish various observability theorems for rectangular membranes by applying Mehrenberger's recent generalization of Ingham's theorem.
Modeling and control of hybrid beam systems with rotating tip component
David L. Russell
2014, 3(2): 305-329 doi: 10.3934/eect.2014.3.305 +[Abstract](36) +[PDF](586.9KB)
We study control and stability for two types of hybrid elastic structures consisting of distributed parameter, beam and rod type, elements coupled at one end to a rotating lumped mass. Applications to control of structural vibrations in wind energy units are indicated but not treated explicitly.
Shape optimization for non-Newtonian fluids in time-dependent domains
Jan Sokołowski and  Jan Stebel
2014, 3(2): 331-348 doi: 10.3934/eect.2014.3.331 +[Abstract](32) +[PDF](449.3KB)
We study the model of an incompressible non-Newtonian fluid in a~moving domain. The domain is defined as a tube built by the velocity field $\mathbf{V}$ and described by the family of domains $\Omega_t$ parametrized by $t\in[0,T]$. A new shape optimization problem associated with the model is defined for a family of initial domains $\Omega_0$ and admissible velocity vector fields. It is shown that such shape optimization problems are well posed under the classical conditions on compactness of the admissible shapes [18]. For the state problem, we prove the existence of weak solutions and their continuity with respect to perturbations of the time-dependent boundary, provided that the power-law index $r\ge11/5$.
Errata: Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping
Lorena Bociu , Petronela Radu and  Daniel Toundykov
2014, 3(2): 349-354 doi: 10.3934/eect.2014.3.349 +[Abstract](36) +[PDF](140.0KB)
This note is an errata for the paper [2] which discusses regular solutions to wave equations with super-critical source terms. The purpose of this note is to address the gap in the proof of uniqueness of such solutions.

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