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Evolution Equations & Control Theory

September 2017 , Volume 6 , Issue 3

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On the Frechet differentiability in optimal control of coefficients in parabolic free boundary problems
Ugur G. Abdulla, Evan Cosgrove and Jonathan Goldfarb
2017, 6(3): 319-344 doi: 10.3934/eect.2017017 +[Abstract](352) +[HTML](21) +[PDF](465.0KB)
Abstract:

We consider the inverse Stefan type free boundary problem, where the coefficients, boundary heat flux, and density of the sources are missing and must be found along with the temperature and the free boundary. We pursue an optimal control framework where boundary heat flux, density of sources, and free boundary are components of the control vector. The optimality criteria consists of the minimization of the $L_2$-norm declinations of the temperature measurements at the final moment, phase transition temperature, and final position of the free boundary. We prove the Frechet differentiability in Besov-Hölder spaces, and derive the formula for the Frechet differential under minimal regularity assumptions on the data. The result implies a necessary condition for optimal control and opens the way to the application of projective gradient methods in Besov-Hölder spaces for the numerical solution of the inverse Stefan problem.

The $\varepsilon$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors
María Anguiano and Alain Haraux
2017, 6(3): 345-356 doi: 10.3934/eect.2017018 +[Abstract](334) +[HTML](28) +[PDF](360.3KB)
Abstract:

We prove an estimation of the Kolmogorov \begin{document} $\varepsilon$ \end{document}-entropy in \begin{document} $H$ \end{document} of the unitary ball in the space \begin{document} $V$ \end{document}, where $H$ is a Hilbert space and \begin{document} $V$ \end{document} is a Sobolev-like subspace of \begin{document} $H$ \end{document}. Then, by means of Zelik's result [7], an estimate of the fractal dimension of the attractors of some nonlinear parabolic equations is established.

Local exact controllability to trajectories of the magneto-micropolar fluid equations
Cung The Anh and Vu Manh Toi
2017, 6(3): 357-379 doi: 10.3934/eect.2017019 +[Abstract](510) +[HTML](35) +[PDF](498.3KB)
Abstract:

In this paper we prove the exact controllability to trajectories of the magneto-micropolar fluid equations with distributed controls. We first establish new Carleman inequalities for the associated linearized system which lead to its null controllability. Then, combining the null controllability of the linearized system with an inverse mapping theorem, we deduce the local exact controllability to trajactories of the nonlinear problem.

Null controllability for parabolic equations with dynamic boundary conditions
Lahcen Maniar, Martin Meyries and Roland Schnaubelt
2017, 6(3): 381-407 doi: 10.3934/eect.2017020 +[Abstract](396) +[HTML](23) +[PDF](571.4KB)
Abstract:

We prove null controllability for linear and semilinear heat equations with dynamic boundary conditions of surface diffusion type. The results are based on a new Carleman estimate for this type of boundary conditions.

$\mathbb{L}^p-$solutions of the stochastic Navier-Stokes equations subject to Lévy noise with $\mathbb{L}^m(\mathbb{R}^m)$ initial data
Manil T. Mohan and Sivaguru S. Sritharan
2017, 6(3): 409-425 doi: 10.3934/eect.2017021 +[Abstract](439) +[HTML](37) +[PDF](439.7KB)
Abstract:

In this work, we establish the local solvability of the stochastic Navier-Stokes equations in \begin{document} $\mathbb{R}^m$ \end{document}, \begin{document} $m≥ 2$ \end{document}, perturbed by Lévy noise in \begin{document} $\mathbb L^p-$ \end{document}spaces for \begin{document} $p∈[m,∞)$ \end{document} with an \begin{document} $\mathbb L^m(\mathbb{R}^m)-$ \end{document}valued initial data.

Sensitivity analysis in set-valued optimization under strictly minimal efficiency
Zhenhua Peng, Zhongping Wan and Weizhi Xiong
2017, 6(3): 427-436 doi: 10.3934/eect.2017022 +[Abstract](356) +[HTML](29) +[PDF](344.5KB)
Abstract:

In this paper, the behavior of the perturbation map is analyzed quantitatively by virtue of contingent derivatives and generalized contingent epiderivatives for the set-valued maps under strictly minimal efficiency. The purpose of this paper is to provide some well-known results concerning sensitivity analysis by applying a separation theorem for convex sets. When the results regress to multiobjective optimization, some related conclusions are obtained in a multiobjective programming problem.

Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping*
Marcio Antonio Jorge da Silva and Vando Narciso
2017, 6(3): 437-470 doi: 10.3934/eect.2017023 +[Abstract](402) +[HTML](28) +[PDF](607.6KB)
Abstract:

In this paper we consider new results on well-posedness and long-time dynamics for a class of extensible beam/plate models whose dissipative effect is given by the product of two nonlinear terms. The addressed model contains a nonlocal nonlinear damping term which generalizes some classes of dissipations usually given in the literature, namely, the linear, the nonlinear and the nonlocal frictional ones. A first mathematical analysis of such damping term is presented and represents the main novelty in our approach.

Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions
Jinrong Wang, Michal Fečkan and Yong Zhou
2017, 6(3): 471-486 doi: 10.3934/eect.2017024 +[Abstract](332) +[HTML](25) +[PDF](399.0KB)
Abstract:

In this paper, we study the approximate controllability of Sobolev-type fractional evolution systems with non-local conditions in Hilbert spaces. Sufficient conditions of approximate controllability of the desired problem are presented by supposing an approximate controllability of the corresponding linear system. By constructing a control function involving Gramian controllability operator, we transform our problem to a fixed point problem of nonlinear operator. Then the Schauder Fixed Point Theorem is applied to complete the proof. An example is given to illustrate our theoretical results.

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