Volumen 27 (2017): Heft 3 (September 2017)
Systems Analysis: Modeling and Control (special section, pp. 457-499), Vyacheslav Maksimov and Boris Mordukhovich (Eds.)

In this paper, we consider the problem of tracking a solution of a reference parabolic equation by a solution of another equation. A stable algorithm based on the extremal shift method is proposed for this problem. The algorithm is designed to work on a sufficiently large time interval where both equations operate.

The paper aims at studying a class of second-order partial differential equations subject to uncertainty involving unknown inputs for which no probabilistic information is available. Developing an approach of feedback control with a model, we derive an efficient reconstruction procedure and thereby design differential equations of reconstruction. A characteristic feature of the obtained equations is that their inputs formed by the feedback control principle constructively approximate unknown inputs of the given second-order distributed parameter system.

This paper describes a numerical scheme for a class of fractional diffusion equations with fixed time delay. The study focuses on the uniqueness, convergence and stability of the resulting numerical solution by means of the discrete energy method. The derivation of a linearized difference scheme with convergence order O(τ^2−α+ h⁴) in L_∞-norm is the main purpose of this study. Numerical experiments are carried out to support the obtained theoretical results.

For abstract linear systems in Hilbert spaces we revisit the problems of exact controllability and complete stabilizability (stabilizability with an arbitrary decay rate), the latter property being related to exact null controllability. We also consider the case when the feedback is not bounded. We obtain a characterization of complete stabilizability for neutral type systems. Conditions for exact null controllability of neutral type systems are discussed. By duality, we obtain a result about continuous final observability. Illustrative examples are given.

This paper is concerned with the controllability of nonlinear fractional delay dynamical systems with implicit fractional derivatives for multiple delays and distributed delays in control variables. Sufficient conditions are obtained by using the Darbo fixed point theorem. Further, examples are given to illustrate the theory.

The aim of this paper is to build and validate a class of energy-preserving schemes for simulating a complex modified Korteweg–de Vries equation. The method is based on a combination of a discrete variational derivative method in time and finite volume element approximation in space. The resulting scheme is accurate, robust and energy-preserving. In addition, for comparison, we also develop a momentum-preserving finite volume element scheme and an implicit midpoint finite volume element scheme. Finally, a complete numerical study is developed to investigate the accuracy, conservation properties and long time behaviors of the energy-preserving scheme, in comparison with the momentum-preserving scheme and the implicit midpoint scheme, for the complex modified Korteweg–de Vries equation.

New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is to get not only an accurate solution, but also an accurate first order derivative at the interface (from each side). The key in 1D is to use the idea of Wheeler (1974). For 2D interface problems, the point is to introduce a small tube near the interface and propose the gradient as part of unknowns, which is similar to a mixed finite element method, but only at the interface. Thus the computational cost is just slightly higher than in the standard finite element method. We present a rigorous one dimensional analysis, which shows a second order convergence order for both the solution and the gradient in 1D. For two dimensional problems, we present numerical results and observe second order convergence for the solution, and super-convergence for the gradient at the interface.

Input-output linearization by state feedback is applied to a flux-controlled active magnetic bearing (AMB) system, operated in the zero-bias mode. Two models of the AMB system are employed. The first one is described by the third-order dynamics with a flux-dependent voltage switching scheme, whereas the second one is the fourth-order system, called self-sensing AMB, since it does not require the measurement of the rotor position. In the case of that system we had to find the flat outputs to guarantee its stability. The proposed control schemes are verified by means of numerical simulations performed within the Matlab environment.

Service systems and their cooperation are one of the most important and hot topics in management and information sciences. To design a reasonable allocation mechanism of service systems is the key issue in the cooperation of service systems. In this paper, we systematically introduce the interval Shapley value as cost allocation of cooperative interval games 〈N, V〉 arising from cooperation in a multi-server service system, and provide an explicit expression for the interval Shapley value of cooperative interval games 〈N, V〉. We construct an interval game 〈N, W〉 of a service system which shares the same value for the grand coalition with the original interval game, by using the characteristic function which is dominated by the function of the original interval game. Finally, we prove that the interval game 〈N, W〉 is concave, which means that the interval Shapley value of the interval game 〈N, W〉 is in the interval core of this interval game, and illustrate this conclusion by using numerical examples.

In a real world situation, whenever ambiguity exists in the modeling of intuitionistic fuzzy numbers (IFNs), interval valued intuitionistic fuzzy numbers (IVIFNs) are often used in order to represent a range of IFNs unstable from the most pessimistic evaluation to the most optimistic one. IVIFNs are a construction which helps us to avoid such a prohibitive complexity. This paper is focused on two types of arithmetic operations on interval valued intuitionistic fuzzy numbers (IVIFNs) to solve the interval valued intuitionistic fuzzy multi-objective linear programming problem with pentagonal intuitionistic fuzzy numbers (PIFNs) by assuming different α and β cut values in a comparative manner. The objective functions involved in the problem are ranked by the ratio ranking method and the problem is solved by the preemptive optimization method. An illustrative example with MATLAB outputs is presented in order to clarify the potential approach.

For many scientists interval arithmetic (IA, I arithmetic) seems to be easy and simple. However, this is not true. Interval arithmetic is complicated. This is confirmed by the fact that, for years, new, alternative versions of this arithmetic have been created and published. These new versions tried to remove shortcomings and weaknesses of previously proposed options of the arithmetic, which decreased the prestige not only of interval arithmetic itself, but also of fuzzy arithmetic, which, to a great extent, is based on it. In our opinion, the main reason for the observed shortcomings of the present IA is the assumption that the direct result of arithmetic operations on intervals is also an interval. However, the interval is not a direct result but only a simplified representative (indicator) of the result. This hypothesis seems surprising, but investigations prove that it is true. The paper shows what conditions should be satisfied by the result of interval arithmetic operations to call it a “result”, how great its dimensionality is, how to perform arithmetic operations and solve equations. Examples illustrate the proposed method of interval computations.

Driven by increasing user requirements and expectations, the fast development of telecommunications networks brings new challenging optimization problems. One of them is routing and spectrum allocation (RSA) of three types of network flows (unicast, anycast, multicast) in elastic optical networks (EONs) implementing dedicated path protection (DPP). In the paper, we model this problem as integer linear programming (ILP) and we introduce two new optimization approaches—a dedicated heuristic algorithm and a column generation (CG)-based method. Then, relying on extensive simulations, we compare algorithm performance with reference methods and evaluate CG efficiency in detail. The results show that the proposed CG method significantly outperforms reference algorithms and achieves results very close to optimal ones (the average distance to optimal results was at most 2.1%).

In recent years elastic optical networks have been perceived as a prospective choice for future optical networks due to better adjustment and utilization of optical resources than is the case with traditional wavelength division multiplexing networks. In the paper we investigate the elastic architecture as the communication network for distributed data centers. We address the problems of optimization of routing and spectrum assignment for large-scale computing systems based on an elastic optical architecture; particularly, we concentrate on anycast user to data center traffic optimization. We assume that computational resources of data centers are limited. For this offline problems we formulate the integer linear programming model and propose a few heuristics, including a meta-heuristic algorithm based on a tabu search method. We report computational results, presenting the quality of approximate solutions and efficiency of the proposed heuristics, and we also analyze and compare some data center allocation scenarios.

Equilibrium analysis in autonomous evolutionary models is of central importance for developing long term treatments. This task typically includes checks on the existence and stability of some equilibria. Prior to touching on the stability, one often attempts to determine the existence where the basic reproductive number ℛ₀ plays a critical role as a threshold parameter. When analyzing a nontrivial equilibrium (e.g., an endemic, boundary, or coexistence equilibrium) where ℛ₀ is explicit, we usually come across a typical result: if ℛ₀ >1, then a nontrivial equilibrium exists in the biological sense. However, for more sophisticated models, ℛ₀ can be too complicated to be revealed in terms of the involving parameters; the task of relating the formulation of a nontrivial equilibrium to ℛ₀ thus becomes intractable. This paper shows how to mitigate such a problem with the aid of functional analysis, adopting the framework of a nonlinear eigenvalue problem. An equilibrium equation is first to be transformed into a canonical equation in a lower dimension, and then the existence is confirmed under several conditions. Three models are tested showing the applicability of this approach.

We introduce a novel concept of the augmented Farey table (AFT). Its purpose is to store the ranks of fractions of a Farey sequence in an efficient manner so as to return the rank of any query fraction in constant time. As a result, computations on the digital plane can be crafted down to simple integer operations; for example, the tasks like determining the extent of collinearity of integer points or of parallelism of straight lines—often required to solve many image-analytic problems—can be made fast and efficient through an appropriate AFT-based tool. We derive certain interesting characterizations of an AFT for its efficient generation. We also show how, for a fraction not present in a Farey sequence, the rank of the nearest fraction in that sequence can efficiently be obtained by the regula falsi method from the AFT concerned. To assert its merit, we show its use in two applications—one in polygonal approximation of digital curves and the other in skew correction of engineering drawings in document images. Experimental results indicate the potential of the AFT in such image-analytic applications.