# American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

All Issues

## Discrete and Continuous Dynamical Systems - S

January 2022 , Volume 15 , Issue 1

Issue on partial differential equations, optimization and their applications

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2022, 15(1): i-ii doi: 10.3934/dcdss.2021163 +[Abstract](5427) +[HTML](136) +[PDF](310.98KB)
Abstract:
2022, 15(1): 1-21 doi: 10.3934/dcdss.2021006 +[Abstract](1581) +[HTML](450) +[PDF](515.54KB)
Abstract:

This work deals with a geometric inverse source problem. It consists in recovering inclusion in a fixed domain based on boundary measurements. The inverse problem is solved via a shape optimization formulation. Two cost functions are investigated, namely, the least squares fitting, and the Kohn-Vogelius function. In this case, the existence of the shape derivative is given via the first order material derivative of the state problems. Furthermore, using the adjoint approach, the shape gradient of both cost functions is characterized. Then, the stability is investigated by proving the compactness of the Hessian at the critical shape for both considered cases. Finally, based on the gradient method, a steepest descent algorithm is developed, and some numerical experiments for non-parametric shapes are discussed.

2022, 15(1): 23-40 doi: 10.3934/dcdss.2021069 +[Abstract](874) +[HTML](285) +[PDF](493.53KB)
Abstract:

In the present work we introduce and study a new method for solving the inverse obstacle problem. It consists in identifying a perfectly conducting inclusion \begin{document}$\omega$\end{document} contained in a larger bounded domain \begin{document}$\Omega$\end{document} via boundary measurements on \begin{document}$\partial \Omega$\end{document}. In order to solve this problem, we use the coupled complex boundary method (CCBM), originaly proposed in [16]. The new method transforms our inverse problem to a complex boundary problem with a complex Robin boundary condition coupling the Dirichlet and Neumann boundary data. Then, we optimize the shape cost function constructed by the imaginary part of the solution in the whole domain in order to determine the inclusion \begin{document}$\omega$\end{document}. Thanks to the tools of shape optimization, we prove the existence of the shape derivative of the complex state with respect to the domain \begin{document}$\omega$\end{document}. We characterize the gradient of the cost functional in order to make a numerical resolution. We then investigate the stability of the optimization problem and explain why this inverse problem is severely ill-posed by proving compactness of the Hessian of cost functional at the critical shape. Finally, some numerical results are presented and compared with classical methods.

2022, 15(1): 41-56 doi: 10.3934/dcdss.2021026 +[Abstract](1173) +[HTML](562) +[PDF](483.55KB)
Abstract:

This paper deals with Sylvester and Stein tensor equations with low rank right hand sides. It proposes extended Krylov-like methods for solving Sylvester and Stein tensor equations. The expressions of residual norms are presented. To show the performance of the proposed approaches, some numerical experiments are given.

2022, 15(1): 57-78 doi: 10.3934/dcdss.2021013 +[Abstract](1371) +[HTML](455) +[PDF](1412.02KB)
Abstract:

Data completion known as Cauchy problem is one most investigated inverse problems. In this work we consider a Cauchy problem associated with Helmholtz equation. Our concerned is the convergence of the well-known alternating iterative method [25]. Our main result is to restore the convergence for the classical iterative algorithm (KMF) when the wave numbers are considerable. This is achieved by, some simple modification for the Neumann condition on the under-specified boundary and replacement by relaxed Neumann ones. Moreover, for the small wave number \begin{document}$k$\end{document}, when the convergence of KMF algorithm's [25] is ensured, our algorithm can be used as an acceleration of convergence.

In this case, we present theoretical results of the convergence of this relaxed algorithm. Meanwhile it, we can deduce the convergence intervals related to the relaxation parameters in different situations. In contrast to the existing results, the proposed algorithm is simple to implement converges for all choice of wave number.

We approach our algorithm using finite element method to obtain an accurate numerical results, to affirm theoretical results and to prove it's effectiveness.

2022, 15(1): 79-93 doi: 10.3934/dcdss.2021030 +[Abstract](1031) +[HTML](374) +[PDF](371.83KB)
Abstract:

We consider a weakly damped cubic nonlinear Schrödinger equation with Dirac interaction defect in a half line of \begin{document}$\mathbb{R}$\end{document}. Endowed with artificial boundary condition at the point \begin{document}$x = 0$\end{document}, we discuss the global existence and uniqueness of solution of this equation by using Faedo–Galerkin method.

2022, 15(1): 95-116 doi: 10.3934/dcdss.2021027 +[Abstract](2263) +[HTML](732) +[PDF](803.35KB)
Abstract:

This paper describes the state of the art and gives a survey of the wide literature published in the last years on the fractional Laplacian. We will first recall some definitions of this operator in \begin{document}$\mathbb{R}^N$\end{document} and its main properties. Then, we will introduce the four main operators often used in the case of a bounded domain; and we will give several simple and significant examples to highlight the difference between these four operators. Also we give a rather long list of references : it is certainly not exhaustive but hopefully rich enough to track most connected results. We hope that this short survey will be useful for young researchers of all ages who wish to have a quick idea of the fractional Laplacian(s).

2022, 15(1): 117-141 doi: 10.3934/dcdss.2021012 +[Abstract](1486) +[HTML](480) +[PDF](455.81KB)
Abstract:

The aim of this paper is to prove existence and regularity of solutions for the following nonlinear singular parabolic problem

Here \begin{document}$\Omega$\end{document} is a bounded open subset of \begin{document}$I\!\!R^{N} (N>p\geq 2), T>0$\end{document} and \begin{document}$f$\end{document} is a non-negative function that belong to some Lebesgue space, \begin{document}$f\in L^{m}(Q)$\end{document}, \begin{document}$Q = \Omega \times(0,T)$\end{document}, \begin{document}$\Gamma = \partial\Omega\times(0,T)$\end{document}, \begin{document}$g(x,t,u) = |u|^{s-1}u$\end{document}, \begin{document}$s\geq 1,$\end{document} \begin{document}$0\leq\theta< 1$\end{document} and \begin{document}$0<\gamma<1.$\end{document}

2022, 15(1): 143-159 doi: 10.3934/dcdss.2021035 +[Abstract](1315) +[HTML](377) +[PDF](527.94KB)
Abstract:

By extending some linear time delay systems stability techniques, this paper, focuses on continuous time delay nonlinear systems (TDNS) dependent delay stability conditions. First, by using the Takagi Sugeno Fuzzy Modeling, a novel relaxed dependent delay stability conditions involving uncommon free matrices, are addressed in Linear Matrix Inequalities (LMI). Then, as application a Nonlinear Carbon Dioxide Model is used and rewritten by a change of coordinate to the interior equilibrium point. Next, by using the non-linearity sector method the model is transformed to a corresponding Fuzzy Takagi Sugeno (TS) multi-model. Also, the maximum delay margin to which the model is stable, is identified. Finally, to prove the analytic results a numerical simulation is also performed and compared to other methods.

2022, 15(1): 161-177 doi: 10.3934/dcdss.2021014 +[Abstract](1069) +[HTML](472) +[PDF](1455.16KB)
Abstract:

We study a simplified multidimensional version of the phenomenological surface growth continuum model derived in [6]. The considered model is a partial differential equation for the surface height profile \begin{document}$u$\end{document} which possesses the following free energy functional:

where \begin{document}$\Omega$\end{document} is the domain of a fixed support on which the growth is carried out. The term \begin{document}$\left|\Delta u \right|^2$\end{document} designates the standard surface diffusion in contrast to the second order term which phenomenologically describes the growth instability. The energy above is mainly carried out in regions of higher surface slope \begin{document}$\left( \left|\nabla u \right| \right)$\end{document}. Hence minimizing such energy corresponds to reducing surface defects during the growth process from a given initial surface configuration. Our analysis is concerned with the energetic and coarsening behaviours of the equilibrium solution. The existence of global energy minimizers and a scaling argument are used to construct a sequence of equilibrium solutions with different wavelength. We apply our minimum energy estimates to derive bounds in terms of the linear system size \begin{document}$\left| \Omega \right|$\end{document} for the characteristic interface width and average slope. We also derive a stable numerical scheme based on the convex-concave decomposition of the energy functional and study its properties while accommodating these results with 1d and 2d numerical simulations.

2022, 15(1): 179-195 doi: 10.3934/dcdss.2021028 +[Abstract](1433) +[HTML](392) +[PDF](443.51KB)
Abstract:

We establish some results regarding the existence of solutions to a nonlinear mono-energetic singular transport equation in slab geometry on \begin{document}$L^p$\end{document}-spaces with \begin{document}$p\in (1,+\infty)$\end{document}. Both the cases where the boundary conditions are specular reflections and periodic are discussed.

2022, 15(1): 197-212 doi: 10.3934/dcdss.2021036 +[Abstract](1652) +[HTML](381) +[PDF](825.59KB)
Abstract:

A methodology for fault-tolerant-control(FTC) is proposed that compensates actuator failures in active suspension systems (ASS). This methodology is based on a Frequency Domain approach that represents failures using a scale factor to optimize the ASS and improve ride comfort. The controller design is carried out using off-the-shelf tools based on linear matrix inequalities (LMIs), guaranteeing asymptotic stability, compensating the effect of actuator faults, and ensuring certain \begin{document}$H_{\infty}$\end{document} performance. In the context of ASS, the performance guarantees correspond to ride comfort in the presence of road disturbances. To validate the approach, controllers are developed and tested in simulation for a quarter-car model: the results illustrate the effectiveness of the proposed approach.

2022, 15(1): 213-227 doi: 10.3934/dcdss.2021015 +[Abstract](1479) +[HTML](456) +[PDF](399.08KB)
Abstract:

In this paper, we establish the existence of weak solution in Orlicz-Sobolev space for the following Kirchhoff type probelm

where \begin{document}$\Omega$\end{document} is a bounded subset in \begin{document}${\mathbb{R}}^N$\end{document}, \begin{document}$N\geq 1$\end{document} with Lipschitz boundary \begin{document}$\partial \Omega.$\end{document} The used technical approach is mainly based on Leray-Shauder's non linear alternative.

2022, 15(1): 229-243 doi: 10.3934/dcdss.2021029 +[Abstract](1204) +[HTML](422) +[PDF](373.55KB)
Abstract:

In this paper, we study the existence of positive solutions of the following equation

The study of the problem \begin{document}$(P_{\lambda})$\end{document} needs generalized Lebesgue and Sobolev spaces. In this work, under suitable assumptions, we prove that some variational methods still work. We use them to prove the existence of positive solutions to the problem \begin{document}$(P_{\lambda})$\end{document} in \begin{document}$W_{0}^{1,p(x)}(\Omega)$\end{document}.

2020 Impact Factor: 2.425
5 Year Impact Factor: 1.490
2021 CiteScore: 3.6

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