Batal, Ahmet
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Batal, A
Batal, A.
Batal, A.
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ahmetbatal@iyte.edu.tr
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04.02. Department of Mathematics
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12
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360
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7
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0
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16
Articles
10
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38994/4569
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2
Supervised PhD Theses
4
WoS Citation Count
64
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61
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4.00
Scopus Citations per Publication
3.81
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10
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6
| Journal | Count |
|---|---|
| Discrete Applied Mathematics | 2 |
| Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics | 1 |
| Electronic Journal of Differential Equations | 1 |
| Evolution Equations & Control Theory | 1 |
| Journal of The Mathematical Society of Japan | 1 |
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Article Citation - WoS: 13Citation - Scopus: 14Nonlinear Schrödinger Equations on the Half-Line With Nonlinear Boundary Conditions(Texas State University - San Marcos, 2016) Batal, Ahmet; Özsarı, TürkerIn this article, we study the initial boundary value problem for nonlinear Schrödinger equations on the half-line with nonlinear boundary conditions ux(0, t) + λ|u(0, t)|ru(0, t) = 0, λ ∈ ℝ − {0}, r > 0. We discuss the local well-posedness when the initial data u0 = u(x, 0) belongs to an L2-based inhomogeneous Sobolev space (formula presented) with (formula presented). We deal with the nonlinear boundary condition by first studying the linear Schrödinger equation with a time-dependent inhomogeneous Neumann boundary condition ux(0, t) = h(t) where (formula presented) (0, T). © 2016 Texas State University.Doctoral Thesis Boundary Feedback Stabilization of Some Evolutionary Partial Differential Equations(01. Izmir Institute of Technology, 2022) Yılmaz, Kemal Cem; Batal, Ahmet; Özsarı, Türker; Özsarı, Türker; Batal, AhmetThe purpose of this study is to control long time behaviour of solutions to some evolutionary partial differential equations posed on a finite interval by backstepping type controllers. At first we consider right endpoint feedback controller design problem for higher-order Schrödinger equation. The second problem is observer design problem, which has particular importance when measurement across the domain is not available. In this case, the sought after right endpoint control inputs involve state of the observer model. However, it is known that classical backstepping strategy fails for designing right endpoint controllers to higher order evolutionary equations. So regarding these controller and observer design problems, we modify the backstepping strategy in such a way that, the zero equilibrium to the associated closed-loop systems become exponentially stable. From the well-posedness point of view, this modification forces us to obtain a time-space regularity estimate which also requires to reveal some smoothing properties for some associated Cauchy problems and an initial-boundary value problem with inhomogeneous boundary conditions. As a third problem, we introduce a finite dimensional version of backstepping controller design for stabilizing infinite dimensional dissipative systems. More precisely, we design a boundary control input involving projection of the state onto a finite dimensional space, which is still capable of stabilizing zero equilibrium to the associated closed-loop system. Our approach is based on defining the backstepping transformation and introducing the associated target model in a novel way, which is inspired from the finite dimensional long time behaviour of dissipative systems. We apply our strategy in the case of reaction-diffusion equation. However, it serves only as a canonical example and our strategy can be applied to various kind of dissipative evolutionary PDEs and system of evolutionary PDEs. We also present several numerical simulations that support our theoretical results.Doctoral Thesis Stability Analysis of Nonlinear Dynamical Systems With Lévy Typeperturbations(01. Izmir Institute of Technology, 2023) Tamcı, Ege; Batal, Ahmet; Savacı, Ferit AcarIn order to model the noise in power networks, generally, normal distribution is used. However, normal distribution is not convenient in modelling noise which has sudden peaks. Instead, combination of a continuous process and a jump processes is much more suitable. With this idea in mind, in this thesis, the stability and control of two equations used in modeling power grids is analyzed, under the assumption that they are exposed to Lévy process noise which includes jumps. These equations are the swing equation and the Kuramoto Model. The swing equation is used to model the single machine infinite bus system (SMIBS). Kuramoto Model is used when a large number of generators are considered as a network of coupled oscillators with their own natural frequencies. In our stability control study in the SMIBS, the noise in the system has sudden and finite changes is assumed and therefore should be modelled with a modified tempered α-stable process obtained by adding a finite jump condition on the tempered α-stable process when α < 1. The control functions depending on the mechanical power input and damping parameters are designed in order to make the system stable in probability and exponential stable at its equilibrium point. These theoretical results are supported by numerical studies. For Kuromato model, assuming that the power network consists of two layers, namely oscillator, and control layers and that is affected with a general Lévy process which has finite jumps, functions which provide the stability of phase and frequencies are obtained, depending on oscillator and coupling strengths. In the light of the numerical studies, the control of frequency and phase synchronization up to a certain noise intensity level can be evaluated, but it is not possible beyond that level is concluded.Article Citation - WoS: 22Citation - Scopus: 20Fokas Method for Linear Boundary Value Problems Involving Mixed Spatial Derivatives(Royal Society of Chemistry, 2020) Fokas, A. S.; Batal, Ahmet; Özsarı, TürkerWe obtain solution representation formulae for some linear initial boundary value problems posed on the half space that involve mixed spatial derivative terms via the unified transform method (UTM), also known as the Fokas method. We first implement the method on the second-order parabolic PDEs; in this case one can alternatively eliminate the mixed derivatives by a linear change of variables. Then, we employ the method to biharmonic problems, where it is not possible to eliminate the cross term via a linear change of variables. A basic ingredient of the UTM is the use of certain invariant maps. It is shown here that these maps are well defined provided that certain analyticity issues are appropriately addressed.Article Parity of an Odd Dominating Set(2022) Batal, AhmetFor a simple graph $G$ with vertex set $V(G)={v_1,...,v_n}$, we define the closed neighborhood set of a vertex $u$ as $N[u]={v in V(G) ; | ; v ; text{is adjacent to} ; u ; text{or} ; v=u }$ and the closed neighborhood matrix $N(G)$ as the matrix whose $i$th column is the characteristic vector of $N[v_i]$. We say a set $S$ is odd dominating if $N[u]cap S$ is odd for all $uin V(G)$. We prove that the parity of the cardinality of an odd dominating set of $G$ is equal to the parity of the rank of $G$, where rank of $G$ is defined as the dimension of the column space of $N(G)$. Using this result we prove several corollaries in one of which we obtain a general formula for the nullity of the join of graphs.Article Citation - WoS: 1Citation - Scopus: 1Irreducibility and Primality in Differentiability Classes(Michigan State University Press, 2023) Batal, Ahmet; Eyidoğan, S.; Göral, HaydarIn this note, we give criteria for the irreducibility of functions in Cm [0, 1], where m ∈ {1, 2, 3, ...} ∪ {∞} ∪ {ω}. We also discuss irreducibility in multivariable differentiability classes. Moreover, we characterize irreducible functions and maximal ideals in C∞ [0, 1]. In fact, irreducible and prime smooth functions are the same, and every maximal ideal of C∞ [0, 1] is principal. © 2023 Michigan State University Press. All rights reserved.Article On the Construction of Xor-Magic Graphs(Elsevier, 2026) Batal, AhmetA simple connected graph of order 2nis defined as a xor-magic graph of power n if its vertices can be labeled with vectors from Fn2 in a one-to-one manner such that the sum of labels in each closed neighborhood set of vertices equals zero. In this paper, we introduce a method called the self-switching operation, which, when properly applied to an odd xor-magic graph of power n, generates a xor-magic graph of power n + 1. We demonstrate the existence of a proper self-switching operation for any given odd xor-magic graph and provide a characterization of the cut space of a connected graph in the process. We also observe that the Dyck graph can be obtained from the complete graph of order 4 by applying three successive self-switching operations. Additionally, we investigate various graph products, including Cartesian, tensor, strong, lexicographical, and modular products. We observe that these products allow us to generate xor-magic graphs by selecting appropriate factor graphs. Notably, we discover that a modular product of graphs is always a xor-magic graph when the orders of its factors are powers of 2 (except for 2 itself). In the process, we realize that the Clebsch graph is the modular product of the cycle graph and the empty graph, each of order 4. By combining the self-switching operation with the modular product, we establish the existence of k-regular xor-magic graphs of power n for all n >= 2 and for all k is an element of {3, 5, 7, ... , 2n-5}boolean OR{2n-1}. We also prove that there is no (2n-3)-regular xor-magic graph of power n. Lastly, we introduce two more methods to produce xor-magic graphs. One method utilizes Cayley graphs and the other utilizes linear algebra. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.Article Citation - WoS: 8Citation - Scopus: 8Stabilization of Higher Order Schrödinger Equations on a Finite Interval: Part I(American Institute of Mathematical Sciences, 2021) Batal, Ahmet; Özsarı, Türker; Yılmaz, Kemal CemWe study the backstepping stabilization of higher order linear and nonlinear Schrödinger equations on a finite interval, where the boundary feedback acts from the left Dirichlet boundary condition. The plant is stabilized with a prescribed rate of decay. The construction of the backstepping kernel is based on a challenging successive approximation analysis. This contrasts with the case of second order pdes. Second, we consider the case where the full state of the system cannot be measured at all times but some partial information such as measurements of a boundary trace are available. For this problem, we simultaneously construct an observer and the associated backstepping controller which is capable of stabilizing the original plant. Wellposedness and regularity results are provided for all pde models. Although the linear part of the model is similar to the KdV equation, the power type nonlinearity brings additional difficulties. We give two examples of boundary conditions and partial measurements. We also present numerical algorithms and simulations verifying our theoretical results to the fullest extent. Our numerical approach is novel in the sense that we solve the target systems first and obtain the solution to the feedback system by using the bounded invertibility of the backstepping transformation. © 2021, American Institute of Mathematical Sciences. All rights reserved.Article Citation - WoS: 7Citation - Scopus: 7Output Feedback Stabilization of the Linearized Korteweg-De Vries Equation With Right Endpoint Controllers(Elsevier Ltd., 2019) Batal, Ahmet; Özsarı, TürkerIn this paper, we prove the output feedback stabilization for the linearized Korteweg-de Vries (KdV) equation posed on a finite domain in the case the full state of the system cannot be measured. We assume that there is a sensor at the left end point of the domain capable of measuring the first and second order boundary traces of the solution. This allows us to design a suitable observer system whose states can be used for constructing boundary feedbacks acting at the right endpoint so that both the observer and the original plant become exponentially stable. Stabilization of the original system is proved in the L-2-sense, while the convergence of the observer system to the original plant is also proved in higher order Sobolev norms. The standard backstepping approach used to construct a left endpoint controller fails and presents mathematical challenges when building right endpoint controllers due to the overdetermined nature of the related kernel models. In order to deal with this difficulty we use the method of Ozsan and Batal, (2019) which is based on using modified target systems involving extra trace terms. In addition, we show that the number of controllers and boundary measurements can be reduced to one, with the cost of a slightly lower exponential rate of decay. We provide numerical simulations illustrating the efficacy of our controllers. (C) 2019 Elsevier Ltd. All rights reserved.Article Citation - WoS: 2Citation - Scopus: 1K-Color Region Select Game(MATH SOC JAPAN, 2023) Batal, Ahmet; Güğümcü, NeslihanThe region select game, introduced by Ayaka Shimizu, Akio Kawauchi and Kengo Kishimoto, is a game that is played on knot diagrams whose crossings are endowed with two colors. The game is based on the region crossing change moves that induce an unknotting operation on knot diagrams. We generalize the region select game to be played on a knot diagram endowed with k-colors at its vertices for 2 <= k <= infinity.