# American Institute of Mathematical Sciences

ISSN:
2155-3289

eISSN:
2155-3297

All Issues

## Numerical Algebra, Control and Optimization

September 2022 , Volume 12 , Issue 3

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2022, 12(3): 469-480 doi: 10.3934/naco.2021015 +[Abstract](1490) +[HTML](596) +[PDF](335.15KB)
Abstract:

In the setting of arbitrary Hilbert spaces, we give a representation of M-P inverse of the sum of linear operators \begin{document}$A+B$\end{document} under suitable conditions. Based on the full-rank decomposition of an operator, we prove that the extension of the Fill-Fishkind formula for \begin{document}$A$\end{document} and \begin{document}$B$\end{document} with closed ranges, remains valid, keeping the same conditions of Fill-Fishkind formula for two matrices, also we obtain an analogous formula under the Fill-Fishkind conditions, beyond we derive some representations of M-P inverse of a 2-by-2 block operator with disjoint ranges.

2022, 12(3): 481-493 doi: 10.3934/naco.2021016 +[Abstract](1386) +[HTML](460) +[PDF](464.95KB)
Abstract:

This paper discusses model order reduction of large sparse second-order index-3 differential algebraic equations (DAEs) by applying Iterative Rational Krylov Algorithm (IRKA). In general, such DAEs arise in constraint mechanics, multibody dynamics, mechatronics and many other branches of sciences and technologies. By deflecting the algebraic equations the second-order index-3 system can be altered into an equivalent standard second-order system. This can be done by projecting the system onto the null space of the constraint matrix. However, creating the projector is computationally expensive and it yields huge bottleneck during the implementation. This paper shows how to find a reduce order model without projecting the system onto the null space of the constraint matrix explicitly. To show the efficiency of the theoretical works we apply them to several data of second-order index-3 models and experimental resultants are discussed in the paper.

2022, 12(3): 495-512 doi: 10.3934/naco.2021018 +[Abstract](1197) +[HTML](493) +[PDF](734.85KB)
Abstract:

This article focuses on the hidden insights about the Rayleigh-Taylor instability of two superimposed horizontal layers of nanofluids having different densities in the presence of rotation factor. Conservation equations are subjected to linear perturbations and further analyzed by using the Normal Mode technique. A dispersion relation incorporating the effects of surface tension, Atwood number, rotation factor and volume fraction of nanoparticles is obtained. Using Routh-Hurtwitz criterion the stable and unstable modes of Rayleigh-Taylor instability are discussed in the presence/absence of nanoparticles and presented through graphs. It is observed that in the absence/presence of nanoparticles, surface tension helps to stabilize the system and Atwood number has a destabilizing impact without the consideration of rotation factor. But if rotation parameter is considered (in the absence/presence of nanoparticles) then surface tension destabilizes the system while Atwood number has a stabilization effect (for a particular range of wave number). The volume fraction of nanoparticles destabilizes the system in the absence of rotation but in the presence of rotation the stability of the system is significantly stimulated by the nanoparticles.

2022, 12(3): 513-535 doi: 10.3934/naco.2021019 +[Abstract](1279) +[HTML](628) +[PDF](5879.03KB)
Abstract:

This article proposes an efficient approach for solving portfolio type problems. It is highly suitable to help fund allocators and decision makers to set up appropriate portfolios for investors. Stock selection is based upon the risk benefits analysis using MADM approach in fuzzy environment. This sort of analysis allows decision makers to identify the list of acceptable portfolios where they can assign some portions of their asset to them. The purpose of this article is two folds; first, to introduce a methodology to select the list of stocks for investment purpose, and second, to employ a stochastic fractional programming model to assign money into selected stocks. This article proposes a hybrid methodology for finding an optimal or new optimal solution of the problem. This hybrid approach considers risks and benefits at the time of stocks prioritization. This is followed by solving a fractional programming to determine the percentages of the budget to be allocated to stocks while dealing with two sets of suitable and non-suitable stocks. For clarification purposes, a sample example problem is solved.

2022, 12(3): 537-550 doi: 10.3934/naco.2021020 +[Abstract](1073) +[HTML](585) +[PDF](344.61KB)
Abstract:

In the present paper, some necessary and su?cient optimality conditions for a convex optimization problem over inequality constraints are presented which are not necessarily convex and are based on convex intersection of non-necessarily convex sets. The oriented distance function and a characterization of the normal cone of the feasible set are used to obtain the optimality conditions. In the second part of the paper, a non-linear smooth optimization model for customer satisfaction in automotive industry is introduced. The results of the first part are applied to solve this problem theoretically.

2022, 12(3): 551-568 doi: 10.3934/naco.2021021 +[Abstract](1175) +[HTML](455) +[PDF](424.11KB)
Abstract:

The current work aims at finding the approximate solution to solve the nonlinear fractional type Volterra integro-differential equation

In order to solve the aforementioned equation, the researchers relied on the Bernstein polynomials besides the fractional Caputo derivatives through applying the collocation method. So, the equation becomes nonlinear system of equations. By solving the former nonlinear system equation, we get the approximate solution in form of Bernstein's fractional series. Besides, we will present some examples with the estimate of the error.

2022, 12(3): 569-582 doi: 10.3934/naco.2021022 +[Abstract](1296) +[HTML](490) +[PDF](931.75KB)
Abstract:

We present an iterative method for solving the convex constraint nonlinear equation problem. The method incorporates the projection strategy by Solodov and Svaiter with the hybrid Liu-Storey and Conjugate descent method by Yang et al. for solving the unconstrained optimization problem. The proposed method does not require the Jacobian information, nor does it require to store any matrix at each iteration. Thus, it has the potential to solve large-scale non-smooth problems. Under some standard assumptions, the convergence analysis of the method is established. Finally, to show the applicability of the proposed method, the proposed method is used to solve the \begin{document}$\ell_1$\end{document}-norm regularized problems to restore blurred and noisy images. The numerical experiment indicates that our result is a significant improvement compared with the related methods for solving the convex constraint nonlinear equation problem.

2022, 12(3): 583-599 doi: 10.3934/naco.2021023 +[Abstract](1056) +[HTML](462) +[PDF](614.84KB)
Abstract:

The route prediction of unmanned aerial vehicles (UAVs) according to their missions is a strategic issue in the aviation field. In some particular missions, the UAV tasks are to start a movement from a defined point to a target reign in the shortest time. This paper proposes a practical method to find the guidance law of the fixed-wing UAV to achieve time-optimal considering the ambient wind. The unique features of this paper are that the environment includes the moving and fixed obstacles as the route constraints, and the fixed-wing UAVs have to keep a given distance from these obstacles. Also, we consider the specific kinematic equation of the fixed-wing UAV and limitations on the flight-path angle and bank-angles as other constraints. We suggest a method for controlling a fixed-wing UAV to get time-optimal using the re-scaling and parameterization techniques. These techniques are useful and effective in maximizing the performance of the gradient-based methods as a sequential quadratic programming method (\begin{document}$SQP$\end{document}) for numerical solutions. Then, all constraints of the time-optimal control problem are converted to a constraint using an exact penalty function. Due to being exact, finding the control variables and switching times is more accurate and faster. Finally, some numerical examples are simulated to explore the effectiveness of our proposed study in reality.

2022, 12(3): 601-610 doi: 10.3934/naco.2021024 +[Abstract](1194) +[HTML](445) +[PDF](448.05KB)
Abstract:

We investigate the consensus stability for linear stochastic multi-agent systems with multiplicative noises and Markovian random graphs and investigate the asymptotic consensus in the mean square sense for the systems. To establish the consensus stability for the systems, we analysis the consensus error systems by developing general stochastic differential equation with jumps, matrix theory and algebraic graph theory, and then show that the error consensus in the mean square sense finally tending to zero as time goes on is determined by the strongly connected property of union of topologies. Finally, we provide an example to demonstrate the effectiveness of our theoretical results.

2022, 12(3): 611-620 doi: 10.3934/naco.2021025 +[Abstract](995) +[HTML](454) +[PDF](963.86KB)
Abstract:

In this manuscript, we design a methodology to investigate the anti-synchronization scheme in chaotic chemical reactor system using adaptive control method (ACM). Initially, an ACM has been proposed and analysed systematically for controlling the microscopic chaos found in the discussed system which is essentially described by employing Lyapunov stability theory (LST). The required asymptotic stability criterion of the state variables of the discussed system having unknown parameters is derived by designing appropriate control functions and parameter updating laws. In addition, numerical simulation results in MATLAB software are performed to illustrate the effective presentation of the considered strategy. Simulations outcomes correspond that the primal aim of chaos control in the given system have been attained computationally.

2022, 12(3): 621-636 doi: 10.3934/naco.2021026 +[Abstract](1655) +[HTML](513) +[PDF](476.8KB)
Abstract:

In this paper, a numerical approximation solution of a space-time fractional diffusion equation (FDE), involving Caputo-Katugampola fractional derivative is considered. Stability and convergence of the proposed scheme are discussed using mathematical induction. Finally, the proposed method is validated through numerical simulation results of different examples.

2022, 12(3): 637-658 doi: 10.3934/naco.2021029 +[Abstract](1235) +[HTML](477) +[PDF](13207.1KB)
Abstract:

In this work a novel mesh adaptation technique is proposed to approximate discontinuous or boundary layer solution of partial differential equations. We introduce new estimator and monitor function to detect solution region containing discontinuity and layered region. Subsequently, this information is utilized along with equi-distribution principle to adapt the mesh locally. Numerical tests for numerous scalar problems are presented. These results clearly demonstrate the robustness of this method and non-oscillatory nature of the computed solutions.

2021 CiteScore: 1.9