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Volume 1, 2012

Evolution Equations & Control Theory

June 2012 , Volume 1 , Issue 1

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Preface
Alain Haraux and Irena Lasiecka
2012, 1(1): i-i doi: 10.3934/eect.2012.1.1i +[Abstract](872) +[PDF](83.9KB)
Abstract:
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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Internal stabilization of Navier-Stokes equation with exact controllability on spaces with finite codimension
Viorel Barbu and Ionuţ Munteanu
2012, 1(1): 1-16 doi: 10.3934/eect.2012.1.1 +[Abstract](763) +[PDF](416.9KB)
Abstract:
One designs an internal stabilizing feedback controller, for the Navier-Stokes equations, which steers, in finite time, the initial value $X_o$ in $X_e+\mathcal{X}_s$, where $X_e$ is any equilibrium solution and $\mathcal{X}_s$ is a finite codimensional space, consisting of stable modes.
On Kelvin-Voigt model and its generalizations
Miroslav Bulíček, Josef Málek and K. R. Rajagopal
2012, 1(1): 17-42 doi: 10.3934/eect.2012.1.17 +[Abstract](1596) +[PDF](596.6KB)
Abstract:
We consider a generalization of the Kelvin-Voigt model where the elastic part of the Cauchy stress depends non-linearly on the linearized strain and the dissipative part of the Cauchy stress is a nonlinear function of the symmetric part of the velocity gradient. The assumption that the Cauchy stress depends non-linearly on the linearized strain can be justified if one starts with the assumption that the kinematical quantity, the left Cauchy-Green stretch tensor, is a nonlinear function of the Cauchy stress, and linearizes under the assumption that the displacement gradient is small. Long-time and large data existence, uniqueness and regularity properties of weak solution to such a generalized Kelvin-Voigt model are established for the non-homogeneous mixed boundary value problem. The main novelty with regard to the mathematical analysis consists in including nonlinear (non-quadratic) dissipation in the problem.
Invariance for stochastic reaction-diffusion equations
Piermarco Cannarsa and Giuseppe Da Prato
2012, 1(1): 43-56 doi: 10.3934/eect.2012.1.43 +[Abstract](769) +[PDF](409.6KB)
Abstract:
Given a stochastic reaction-diffusion equation on a bounded open subset $\mathcal O$ of $\mathbb{R}^n$, we discuss conditions for the invariance of a nonempty closed convex subset $K$ of $L^2(\mathcal O)$ under the corresponding flow. We obtain two general results under the assumption that the fourth power of the distance from $K$ is of class $C^2$, providing, respectively, a necessary and a sufficient condition for invariance. We also study the example where $K$ is the cone of all nonnegative functions in $L^2(\mathcal O)$.
Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations
Igor Chueshov and Alexey Shcherbina
2012, 1(1): 57-80 doi: 10.3934/eect.2012.1.57 +[Abstract](835) +[PDF](528.6KB)
Abstract:
We deal with an initial boundary value problem for the Schrödinger-Boussinesq system arising in plasma physics in two-dimensional domains. We prove the global Hadamard well-posedness of this problem (with respect to the topology which is weaker than topology associated with the standard variational (weak) solutions) and study properties of the solutions. In the dissipative case the existence of a global attractor is established.
Optimal control of advective direction in reaction-diffusion population models
Heather Finotti, Suzanne Lenhart and Tuoc Van Phan
2012, 1(1): 81-107 doi: 10.3934/eect.2012.1.81 +[Abstract](1017) +[PDF](809.3KB)
Abstract:
We investigate optimal control of the advective coefficient in a class of parabolic partial differential equations, modeling a population with nonlinear growth. This work is motivated by the question: Does movement toward a better resource environment benefit a population? Our objective functional is formulated with interpreting "benefit" as the total population size integrated over our finite time interval. Results on existence, uniqueness, and characterization of the optimal control are established. Our numerical illustrations for several growth functions and resource functions indicate that movement along the resource spatial gradient benefits the population, meaning that the optimal control is close to the spatial gradient of the resource function.
Certain questions of feedback stabilization for Navier-Stokes equations
Andrei Fursikov and Alexey V. Gorshkov
2012, 1(1): 109-140 doi: 10.3934/eect.2012.1.109 +[Abstract](885) +[PDF](624.3KB)
Abstract:
The authors study the stabilization problem for Navier-Stokes and Oseen equations near steady-state solution by feedback control. The cases of control in initial condition (start control) as well as impulse and distributed controls in right side supported in a fixed subdomain of the domain $G$ filled with a fluid are investigated. The cases of bounded and unbounded domain $G$ are considered.
Carleman estimates for some anisotropic elasticity systems and applications
Victor Isakov
2012, 1(1): 141-154 doi: 10.3934/eect.2012.1.141 +[Abstract](739) +[PDF](366.7KB)
Abstract:
We show that under some conditions one can obtain Carleman type estimates for the transversely isotropic elasticity system with residual stress. We consider both time dependent and static cases. The main idea is to reduce this system to a principally upper triangular one and the main technical tool is Carleman estimates with two large parameters for general second order partial differential operators.
Modeling of a nonlinear plate
Shun Li and Peng-Fei Yao
2012, 1(1): 155-169 doi: 10.3934/eect.2012.1.155 +[Abstract](864) +[PDF](386.9KB)
Abstract:
We consider modeling of a nonlinear thin plate under the following assumptions: (a) the materials are nonlinear; (b) the deflections are small (linear strain displacement relations). When the middle surface is planar, we consider the bending of a plate to establish the strain energy, the equilibrium equations, and the motion equations. For a shell with a curved middle surface in $\mathbb{R}^3$, we derive a nonlinear model where a deformation in three-dimensions is concerned.
On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities
Jan Prüss, Yoshihiro Shibata, Senjo Shimizu and Gieri Simonett
2012, 1(1): 171-194 doi: 10.3934/eect.2012.1.171 +[Abstract](1124) +[PDF](459.7KB)
Abstract:
The basic model for incompressible two-phase flows with phase transitions is derived from basic principles and shown to be thermodynamically consistent in the sense that the total energy is conserved and the total entropy is nondecreasing. The local well-posedness of such problems is proved by means of the technique of maximal $L_p$-regularity in the case of equal densities. This way we obtain a local semiflow on a well-defined nonlinear state manifold. The equilibria of the system in absence of external forces are identified and it is shown that the negative total entropy is a strict Ljapunov functional for the system. If a solution does not develop singularities, it is proved that it exists globally in time, its orbit is relatively compact, and its limit set is nonempty and contained in the set of equilibria.
Hyperbolic Navier-Stokes equations I: Local well-posedness
Reinhard Racke and Jürgen Saal
2012, 1(1): 195-215 doi: 10.3934/eect.2012.1.195 +[Abstract](1024) +[PDF](426.3KB)
Abstract:
We replace a Fourier type law by a Cattaneo type law in the derivation of the fundamental equations of fluid mechanics. This leads to hyperbolicly perturbed quasilinear Navier-Stokes equations. For this problem the standard approach by means of quasilinear symmetric hyperbolic systems seems to fail by the fact that finite propagation speed might not be expected. Therefore a somewhat different approach via viscosity solutions is developed in order to prove higher regularity energy estimates for the linearized system. Surprisingly, this method yields stronger results than previous methods, by the fact that we can relax the regularity assumptions on the coefficients to a minimum. This leads to a short and elegant proof of a local-in-time existence result for the corresponding first order quasilinear system, hence also for the original hyperbolicly perturbed Navier-Stokes equations.
Hyperbolic Navier-Stokes equations II: Global existence of small solutions
Reinhard Racke and Jürgen Saal
2012, 1(1): 217-234 doi: 10.3934/eect.2012.1.217 +[Abstract](959) +[PDF](425.9KB)
Abstract:
We consider a hyperbolicly perturbed Navier-Stokes initial value problem in ${\mathbb R}^n$, $n=2,3$, arising from using a Cattaneo type relation instead of a Fourier type one in the constitutive equations. The resulting system is an essentially hyperbolic one with quasilinear nonlinearities. The global existence of smooth solutions for small data is proved, and relations to the classical Navier-Stokes systems are discussed.

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