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Evolution Equations and Control Theory

Open Access Articles

A shape optimization problem constrained with the Stokes equations to address maximization of vortices
John Sebastian Simon and Hirofumi Notsu
2022 doi: 10.3934/eect.2022003 +[Abstract](539) +[HTML](140) +[PDF](5728.49KB)

We study an optimization problem that aims to determine the shape of an obstacle that is submerged in a fluid governed by the Stokes equations. The mentioned flow takes place in a channel, which motivated the imposition of a Poiseuille-like input function on one end and a do-nothing boundary condition on the other. The maximization of the vorticity is addressed by the \begin{document}$ L^2 $\end{document}-norm of the curl and the det-grad measure of the fluid. We impose a Tikhonov regularization in the form of a perimeter functional and a volume constraint to address the possibility of topological change. Having been able to establish the existence of an optimal shape, the first order necessary condition was formulated by utilizing the so-called rearrangement method. Finally, numerical examples are presented by utilizing a finite element method on the governing states, and a gradient descent method for the deformation of the domain. On the said gradient descent method, we use two approaches to address the volume constraint: one is by utilizing the augmented Lagrangian method; and the other one is by utilizing a class of divergence-free deformation fields.

Well-posedness of linear first order port-Hamiltonian Systems on multidimensional spatial domains
Nathanael Skrepek
2021, 10(4): 965-1006 doi: 10.3934/eect.2020098 +[Abstract](1267) +[HTML](691) +[PDF](587.15KB)

We consider a port-Hamiltonian system on an open spatial domain \begin{document}$ \Omega \subseteq \mathbb{R}^n $\end{document} with bounded Lipschitz boundary. We show that there is a boundary triple associated to this system. Hence, we can characterize all boundary conditions that provide unique solutions that are non-increasing in the Hamiltonian. As a by-product we develop the theory of quasi Gelfand triples. Adding "natural" boundary controls and boundary observations yields scattering/impedance passive boundary control systems. This framework will be applied to the wave equation, Maxwell's equations and Mindlin plate model. Probably, there are even more applications.

Weimin Han, Stanislaw Migórski and Mircea Sofonea
2020, 9(4): i-ii doi: 10.3934/eect.2020090 +[Abstract](992) +[HTML](832) +[PDF](13635.75KB)
Introduction to the special issue "Nonlinear wave phenomena in continuum physics: Some recent findings"
Pedro M. Jordan and Barbara Kaltenbacher
2019, 8(1): i-iii doi: 10.3934/eect.20191i +[Abstract](3345) +[HTML](772) +[PDF](152.61KB)
Front matter
Marcelo Disconzi, Daniel Toundykov and Justin T. Webster
2016, 5(4): i-iii doi: 10.3934/eect.201604i +[Abstract](2474) +[PDF](110.5KB)
This special volume of Evolution Equations and Control Theory commemorates the results presented at a mini-symposium on ``Analysis and Control of Fluid Models and Flow-coupled Systems" in December 2015. This meeting was part of the SIAM Conference on Analysis of Partial Differential Equations, held December 7--10, 2015 in Scottsdale, Arizona at the DoubleTree Resort by Hilton in Paradise Valley. The mini-symposium was organized by the Editors: Marcelo Disconzi, Irena Lasiecka, Daniel Toundykov, and Justin Webster.

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Introduction to the special volume ``Mathematics of nonlinear acoustics: New approaches in analysis and modeling''
Pedro M. Jordan and Barbara Kaltenbacher
2016, 5(3): i-ii doi: 10.3934/eect.201603i +[Abstract](2982) +[PDF](107.1KB)
Over the last 12--15 years, there has been a resurgence of interest in the study of nonlinear acoustic phenomena. Using the tools of both classical mathematical analysis and computational physics, researchers have obtained a wide range of new results, some of which might be described as remarkable. As with almost all trends in science, the reasons for this revival are varied: they range from practical applications (e.g., the need to improve our understanding of high-intensity ultrasound); to the development of numerical schemes which are better at capturing the physics of nonlinear compressible flow; to new acoustic models which lend themselves to study by analytical methods.

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Mathematics of nonlinear acoustics
Barbara Kaltenbacher
2015, 4(4): 447-491 doi: 10.3934/eect.2015.4.447 +[Abstract](6684) +[PDF](724.1KB)
The aim of this paper is to highlight some recent developments and outcomes in the mathematical analysis of partial differential equations describing nonlinear sound propagation. Here the emphasis lies on well-posedness and decay results, first of all for the classical models of nonlinear acoustics, later on also for some higher order models. Besides quoting results, we also try to give an idea on their derivation by showning some of the crucial energy estimates. A section is devoted to optimization problems arising in the practical use of high intensity ultrasound.
    While this review puts a certain focus on results obtained in the context of the mentioned FWF project, we also provide some important additional references (although certainly not all of them) for interesting further reading.
Thomas Seidman, Samira El Yacoubi and Abdessamad Tridane
2015, 4(2): i-iv doi: 10.3934/eect.2015.4.2i +[Abstract](2856) +[PDF](141.4KB)
This special issue of the Journal of Evolution Equations and Control Theory (EECT) is dedicated to Professor Abdelhaq El Jai on the occasion of his retirement and in celebration of his significant achievements in the field of control and distributed parameter systems theory, both in his own research and in his leadership in the development of a Moroccan research community in DPS.

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Claudio Giorgi, Davide Guidetti and Maria Grazia Naso
2014, 3(3): i-ii doi: 10.3934/eect.2014.3.3i +[Abstract](2774) +[PDF](7910.0KB)
1.Mauro Fabrizio. This volume entitled ``Mathematical Models and Analytical Problems in Modern Continuum Thermomechanics" is dedicated to Mauro Fabrizio on the occasion of his retirement.

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Stephen W. Taylor and Roberto Triggiani
2013, 2(4): i-ii doi: 10.3934/eect.2013.2.4i +[Abstract](2745) +[PDF](305.1KB)
This volume is dedicated to Walter Littman on the occasion of his retirement.

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Preface: Introduction to the Special Volume on Nonlinear PDEs and Control Theory with Applications
Lorena Bociu, Barbara Kaltenbacher and Petronela Radu
2013, 2(2): i-ii doi: 10.3934/eect.2013.2.2i +[Abstract](3011) +[PDF](93.4KB)
This volume collects a number of contributions in the fields of partial differential equations and control theory, following the Special Session Nonlinear PDEs and Control Theory with Applications held at the 9th AIMS conference on Dynamical Systems, Differential Equations and Applications in Orlando, July 1--5, 2012.

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Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise
Kumarasamy Sakthivel and Sivaguru S. Sritharan
2012, 1(2): 355-392 doi: 10.3934/eect.2012.1.355 +[Abstract](5265) +[PDF](583.6KB)
In this paper, we establish the solvability of martingale solutions for the stochastic Navier-Stokes equations with Itô-Lévy noise in bounded and unbounded domains in $ \mathbb{R} ^d$,$d=2,3.$ The tightness criteria for the laws of a sequence of semimartingales is obtained from a theorem of Rebolledo as formulated by Metivier for the Lusin space valued processes. The existence of martingale solutions (in the sense of Stroock and Varadhan) relies on a generalization of Minty-Browder technique to stochastic case obtained from the local monotonicity of the drift term.
Alain Haraux and Irena Lasiecka
2012, 1(1): i-i doi: 10.3934/eect.2012.1.1i +[Abstract](3236) +[PDF](83.9KB)
The present Inaugural Volume is the first Issue of a new journal Evolution Equations and Control Theory [EECT], which is published within the AIMS Series. EECT is devoted to topics lying at the interface between Evolution Equations and Control Theory of Dynamics. Evolution equations are to be understood in a broad sense as Infinite Dimensional Dynamics which often arise in modeling physical systems as an infinite-dimensional process. This includes single PDE (Partial Differential Equations) or FDE (Functional Differential Equations) as well as coupled dynamics of different characteristics with an interface between them. Since modern control theory intrinsically depends on a good understanding of the qualitative theory of dynamics and evolution theory, the choice of these two topics appears synergistic and most natural. Past experience shows that new developments in control theory often depend on sufficient information related to the associated dynamical properties of the system. On the other hand, developments in evolution theory allow one to consider certain control theoretic formulations that alone would not appear treatable.

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2020 Impact Factor: 1.081
5 Year Impact Factor: 1.269
2020 CiteScore: 1.6



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