# American Institute of Mathematical Sciences

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2163-2480

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## Evolution Equations & Control Theory

March 2021 , Volume 10 , Issue 1

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2021, 10(1): 1-23 doi: 10.3934/eect.2020039 +[Abstract](867) +[HTML](391) +[PDF](483.18KB)
Abstract:

We consider a three-dimensional Navier-Stokes-Voigt equations with memory in lacking instantaneous kinematic viscosity, in presence of Ekman type damping and singularly oscillating external forces depending on a positive parameter \begin{document}$\varepsilon$\end{document}. Under suitable assumptions on the memory term and on the external forces, we prove the existence and the uniform (w.r.t. \begin{document}$\varepsilon$\end{document}) boundedness as well as the convergence as \begin{document}$\varepsilon$\end{document} tends to \begin{document}$0$\end{document} of uniform attractors \begin{document}$\mathcal A ^\varepsilon$\end{document} of a family of processes associated to the model.

2021, 10(1): 25-36 doi: 10.3934/eect.2020040 +[Abstract](847) +[HTML](391) +[PDF](412.8KB)
Abstract:

This paper focuses on the global existence for a class of Keller-Segel models with signal-dependent motility and general logistic term under homogeneous Neumann boundary conditions in a two-dimensional smoothly bounded domain. We show that if \begin{document}$\lambda\in\mathbb{R}, \, \mu>0$\end{document} and \begin{document}$l>2$\end{document} are constants, then for all sufficiently smooth initial data the system

possesses a global classical solution.

2021, 10(1): 37-59 doi: 10.3934/eect.2020051 +[Abstract](633) +[HTML](284) +[PDF](447.79KB)
Abstract:

The paper deals with the optimal control problem described by second order evolution differential inclusions; to this end first we use an auxiliary problem with second order discrete and discrete-approximate inclusions. Then applying infimal convolution concept of convex functions, step by step we construct the dual problems for discrete, discrete-approximate and differential inclusions and prove duality results. It seems that the Euler-Lagrange type inclusions are "duality relations" for both primary and dual problems and that the dual problem for discrete-approximate problem make a bridge between them. At the end of the paper duality in problems with second order linear discrete and continuous models and model of control problem with polyhedral DFIs are considered.

2021, 10(1): 61-102 doi: 10.3934/eect.2020052 +[Abstract](891) +[HTML](304) +[PDF](632.1KB)
Abstract:

The main goal of this paper is to investigate the boundary controllability of some coupled parabolic systems in the cascade form in the case where the boundary conditions are of Robin type. In particular, we prove that the associated controls satisfy suitable uniform bounds with respect to the Robin parameters, that let us show that they converge towards a Dirichlet control when the Robin parameters go to infinity. This is a justification of the popular penalisation method for dealing with Dirichlet boundary data in the framework of the controllability of coupled parabolic systems.

2021, 10(1): 103-127 doi: 10.3934/eect.2020053 +[Abstract](742) +[HTML](269) +[PDF](554.04KB)
Abstract:

This paper deals with the problem of finding the function \begin{document}$u(x,t)$\end{document}, \begin{document}$(x,t)\in \Omega \times [0,T]$\end{document}, from the final data \begin{document}$u(x,T) = g(x)$\end{document} and \begin{document}$u_t(x,T) = {h(x)}$\end{document},

This problem is known as the inverse initial problem for the nonlinear hyperbolic equation with damping term and it is ill-posed in the sense of Hadamard. In order to stabilize the solution, we propose the filter regularization method to regularize the solution. We establish appropriate filtering functions in cases where the nonlinear source \begin{document}$\mathcal R$\end{document} satisfies the global Lipschitz condition and the specific case \begin{document}$\mathcal R(u) = u|u|^{p-1}, p>1$\end{document} which satisfies the local Lipschitz condition. In addition, we show that regularized solutions converge to the sought solution under a priori assumptions in Gevrey spaces.

2021, 10(1): 129-153 doi: 10.3934/eect.2020054 +[Abstract](618) +[HTML](274) +[PDF](694.74KB)
Abstract:

We study the boundary observability of the 1-D homogeneous wave equation when using a Legendre-Galerkin semi-discretization method. It is already known that spurious high frequencies are responsible for its lack of uniformity with respect to the discretization parameter [4] which may prevent convergence in the approximation of the associated controllability problem. A classical remedy is to filter out the highest frequency components but this comes with a high computational cost in several space dimensions. We present here three remedies: a spectral filtering method, a mixed formulation (already used in the context of finite element method [14]) and a Nitsche's method. Our numerical results show that the uniform boundary observability inequalities are recovered. On the other hand, surprisingly, none of them seem to provide the trace (or direct) inequality uniformly, a property used to prove the convergence of the numerical controls [11]. However, our numerical tests suggest that convergence of the numerical controls is ensured when the uniform observability inequality holds.

2021, 10(1): 155-198 doi: 10.3934/eect.2020061 +[Abstract](685) +[HTML](250) +[PDF](717.87KB)
Abstract:

In this article we provide a local wellposedness theory for quasilinear Maxwell equations with absorbing boundary conditions in \begin{document}${\mathcal{H}}^m$\end{document} for \begin{document}$m \geq 3$\end{document}. The Maxwell equations are equipped with instantaneous nonlinear material laws leading to a quasilinear symmetric hyperbolic first order system. We consider both linear and nonlinear absorbing boundary conditions. We show existence and uniqueness of a local solution, provide a blow-up criterion in the Lipschitz norm, and prove the continuous dependence on the data. In the case of nonlinear boundary conditions we need a smallness assumption on the tangential trace of the solution. The proof is based on detailed apriori estimates and the regularity theory for the corresponding linear problem which we also develop here.

2021, 10(1): 199-227 doi: 10.3934/eect.2020062 +[Abstract](553) +[HTML](253) +[PDF](2798.91KB)
Abstract:

An explicit saturating set consisting of eigenfunctions of Stokes operator in general 3D Cylinders is proposed. The existence of saturating sets implies the approximate controllability for Navier–Stokes equations in \begin{document}$\rm 3D$\end{document} Cylinders under Lions boundary conditions.

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