Nonlinear Analysis: Modelling and Control
http://www.journals.vu.lt/nonlinear-analysis
<p>Founded in 1997. Journal provides a multidisciplinary forum for scientists, researchers and engineers involved in research and design of nonlinear processes and phenomena, including the nonlinear modelling of phenomena of the nature. </p>Vilnius University Pressen-USNonlinear Analysis: Modelling and Control1392-5113<p>Please read the Copyright Notice in <a href="http://www.zurnalai.vu.lt/acta-paedagogica-vilnensia/journalpolicy">Journal Policy</a>. </p>Modelling and simulation of mitral valve for transapical repair applications
http://www.journals.vu.lt/nonlinear-analysis/article/view/12956
<p>Development and application of the numerical model for the simulation of human heart mitral valve (MV) transapical repair is presented. Transapical repair with neochordae implantation is a novel surgical technique allowing beating-heart correction of mitral regurgitation caused by chordae tendineae rupture through a minimally-invasive approach. In the present study, the structural finite element model of the MV decoupled from the blood flow is considered. It comprises two leaflets and chordae tendineae described by nonlinear material model. Geometry of the model and kinematic boundary conditions for fixed points of MV annulus, papillary muscles, and left ventricle apex are defined by patient-specific data. Decoupled behavior of blood is specified by the time-dependent physiologic transvalvular pressure.</p> <p>Personalized computational modelling strategy is applied to perform virtual transapical MV repair by positioning neochordae following the real-life surgery procedure executed by surgeons. A transient analysis in time frame between end-diastole and peak systole is conducted to evaluate post-repair MV function. Computational MV simulation and modelling results provide quantitative information about the neochordae contribution to the MV function improvement and present practical value for the surgical planning of transapical MV repair.</p> <p> </p>Gediminas GaidulisMatteo SelmiDiana ZakarkaitėAudrius AidietisRimantas Kačianauskas
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2019-06-272019-06-27244485–502485–50210.15388/NA.2019.4.1Solvability of control problem for fractional nonlinear differential inclusions with nonlocal conditions
http://www.journals.vu.lt/nonlinear-analysis/article/view/12957
<p>In this paper, we study the approximate controllability of nonlocal fractional differential inclusions involving the Caputo fractional derivative of order <em>q</em> ∈ (1,2) in a Hilbert space. Utilizing measure of noncompactness and multivalued fixed point strategy, a new set of sufficient conditions is obtained to ensure the approximate controllability of nonlocal fractional differential inclusions when the multivalued maps are convex. Precisely, the results are developed under the assumption that the corresponding linear system is approximately controllable.</p> <p> </p> <p> </p>Alka ChadhaRathinasamy SakthivelSwaroop Nandan Bora
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2019-06-272019-06-27244503–522503–52210.15388/NA.2019.4.2Asymptotic behaviour of mild solution of nonlinear stochastic partial functional equations
http://www.journals.vu.lt/nonlinear-analysis/article/view/12968
<p>This paper presents conditions to assure existence, uniqueness and stability for impulsive neutral stochastic integrodifferential equations with delay driven by Rosenblatt process and Poisson jumps. The Banach fixed point theorem and the theory of resolvent operator developed by Grimmer [R.C. Grimmer, Resolvent operators for integral equations in a Banach space, <em>Trans. Am. Math. Soc.</em>, 273(1):333–349, 1982] are used. An example illustrates the potential benefits of these results.</p>Mamadou Abdoul DiopAmour Toffodji Gbaguidi AmoussouCarlos OgouyandjouRathinasamy Sakthivel
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2019-06-272019-06-27244523–544523–54410.15388/NA.2019.4.3Trajectory tracking control of a quadrotor UAV based on sliding mode active disturbance rejection control
http://www.journals.vu.lt/nonlinear-analysis/article/view/12961
<p>This paper proposes a sliding mode active disturbance rejection control scheme to deal with trajectory tracking control problems for the quadrotor unmanned aerial vehicle (UAV). Firstly, the differential signal of the reference trajectory can be obtained directly by using the tracking differentiator (TD), then the design processes of the controller can be simplified. Secondly, the estimated values of the UAV's velocities, angular velocities, total disturbance can be acquired by using extended state observer (ESO), and the total disturbance of the system can be compensated in the controller in real time, then the robustness and anti-interference capability of the system can be improved. Finally, the sliding mode controller based on TD and ESO is designed, the stability of the closed-loop system is proved by Lyapunov method. Simulation results show that the control scheme proposed in this paper can make the quadrotor track the desired trajectory quickly and accurately.</p> <p> </p>Yong ZhangZengqiang ChenMingwei SunXinghui Zhang
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2019-06-272019-06-27244545–560545–56010.15388/NA.2019.4.4Dissipativity analysis of stochastic fuzzy neural networks with randomly occurring uncertainties using delay dividing approach
http://www.journals.vu.lt/nonlinear-analysis/article/view/12962
<p>This paper focuses on the problem of delay-dependent robust dissipativity analysis for a class of stochastic fuzzy neural networks with time-varying delay. The randomly occurring uncertainties under consideration are assumed to follow certain mutually uncorrelated Bernoulli-distributed white noise sequences. Based on the Itô's differential formula, Lyapunov stability theory, and linear matrix inequalities techniques, several novel sufficient conditions are derived using delay partitioning approach to ensure the dissipativity of neural networks with or without time-varying parametric uncertainties. It is shown, by comparing with existing approaches, that the delay-partitioning projection approach can largely reduce the conservatism of the stability results. Numerical examples are constructed to show the effectiveness of the theoretical results.</p>Sundaram SenthilrajRamachandran RajaJinde CaoHabib M. Fardoun
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2019-06-272019-06-27244561–581561–58110.15388/NA.2019.4.5Nonlocal q-fractional boundary value problem with Stieltjes integral conditions
http://www.journals.vu.lt/nonlinear-analysis/article/view/12963
<p>In this paper, we are dedicated to investigating a new class of one-dimensional lower-order fractional <em>q</em>-differential equations involving integral boundary conditions supplemented with Stieltjes integral. This condition is more general as it contains an arbitrary order derivative. It should be pointed out that the problem discussed in the current setting provides further insight into the research on nonlocal and integral boundary value problems. We first give the Green's functions of the boundary value problem and then develop some properties of the Green's functions that are conductive to our main results. Our main aim is to present two results: one considering the uniqueness of nontrivial solutions is given by virtue of contraction mapping principle associated with properties of <em>u</em><sub>0</sub>-positive linear operator in which Lipschitz constant is associated with the first eigenvalue corresponding to related linear operator, while the other one aims to obtain the existence of multiple positive solutions under some appropriate conditions via standard fixed point theorems due to Krasnoselskii and Leggett–Williams. Finally, we give an example to illustrate the main results.</p> <p> </p>Jing RenChengbo Zhai
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2019-06-272019-06-27244582–602582–60210.15388/NA.2019.4.6Robust fixed-time synchronization of discontinuous Cohen–Grossberg neural networks with mixed time delays
http://www.journals.vu.lt/nonlinear-analysis/article/view/12964
<p>This paper aims to investigate the fixed-time synchronization (i.e., synchronization in fixed-time sense) of Cohen–Grossberg drive-response neural networks with discontinuous neuron activations and mixed time delays (both time-varying discrete delay and distributed delay). To accomplish the target of fixed-time synchronization, a novel discontinuous feedback control procedure is firstly designed for the response neural networks. Then, under the framework of Filippov solutions, by means of functional differential inclusions theory, inequality technique and the nonsmooth analysis theory with Lyapunov-like approach, some sufficient criteria are derived to design the control parameters for achieving fixed-time synchronization of the proposed drive-response systems. Finally, two numerical examples are presented to illustrate the proposed methodologies.</p>Fanchao KongQuanxin ZhuFeng LiangJuan J. Nieto
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2019-06-272019-06-27244603–625603–62510.15388/NA.2019.4.7Numerical algorithms for solving the optimal control problem of simple bioreactors
http://www.journals.vu.lt/nonlinear-analysis/article/view/12965
<p>The modified nonlocal feedback controller is used to control the production of drugs in a simple bioreactor. This bioreactor is based on the enzymatic conversion of substrate into the required product. The dynamics of this device is described by a system of two nonstationary nonlinear diffusion–convection–reaction equations. The analysis of the influence of the convection transport is one the aims of this paper. The control loop is defined using the relation, which shows how the amount of the drug produced in the bioreactor and delivered into a human body depends on the substrate concentration specified on the external boundary of the bioreactor. The system of PDEs is solved by using the finite volume and finite difference methods, the control loop parameters are defined from the analysis of stationary linearized equations. The second aim of this paper is to solve the inverse problem and to determine optimal boundary conditions. These results enable us to estimate the potential accuracy of the proposed devices.</p> <p> </p>Raimondas ČiegisRemigijus Čiegis
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2019-06-272019-06-27244626–638626–63810.15388/NA.2019.4.8Distance between the fractional Brownian motion and the space of adapted Gaussian martingales
http://www.journals.vu.lt/nonlinear-analysis/article/view/12966
<p>We consider the distance between the fractional Brownian motion defined on the interval [0,1] and the space of Gaussian martingales adapted to the same filtration. As the distance between stochastic processes, we take the maximum over [0,1] of mean-square deviances between the values of the processes. The aim is to calculate the function <em>a</em> in the Gaussian martingale representation ∫<sub>0</sub><em><sup>t</sup>a</em>(<em>s</em>)d<em>W<sub>s</sub></em> that minimizes this distance. So, we have the minimax problem that is solved by the methods of convex analysis. Since the minimizing function <em>a</em> can not be either presented analytically or calculated explicitly, we perform discretization of the problem and evaluate the discretized version of the function <em>a</em> numerically.</p>Yuliya MishuraSergiy Shklyar
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2019-06-272019-06-27244639–657639–65710.15388/NA.2019.4.9Existence of positive solutions for a semipositone discrete boundary value problem
http://www.journals.vu.lt/nonlinear-analysis/article/view/12967
<p>We investigate the existence of positive solutions for a nonlinear second-order difference equation with a linear term and a sign-changing nonlinearity, supplemented with multi-point boundary conditions. In the proof of our main results, we use the Guo–Krasnosel'skii fixed point theorem.</p> <p> </p>Rodica Luca
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2019-06-272019-06-27244658–678658–67810.15388/NA.2019.4.10