<abstract>
<title style='display:none'>Abstract</title>
In this paper, experimental data, given in the form of pairwise comparisons, such as distances or similarities, are considered. Clustering algorithms for processing such data are developed based on the well-known k-means procedure. Relations to factor analysis are shown. The problems of improving clustering quality and of finding the proper number of clusters in the case of pairwise comparisons are considered. Illustrative examples are provided.
</abstract>

Clustering of Data Represented by Pairwise Comparisons

<abstract>
<title style='display:none'>Abstract</title>
In this paper, we introduce a <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_candc-2021-0017_eq_001.png"/>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>
<tex-math>Q_2^n\left( x \right)</tex-math>
</alternatives>
</inline-formula> matrix, whose elements are balancing polynomials, and develop a new coding and decoding method following from the <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_candc-2021-0017_eq_002.png"/>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>
<tex-math>Q_2^n\left( x \right)</tex-math>
</alternatives>
</inline-formula> matrix. We establish the relations between the code matrix elements, error detection and correction for this coding theory.
</abstract>

Coding theory based on balancing polynomials

<abstract>
<title style='display:none'>Abstract</title>
This paper is devoted to a sufficient second-order condition for a weak local minimum in a simple optimal control problem with one control constraint <italic>G</italic>(<italic>u</italic>) ≤ 0, given by a <italic>C</italic>2-function. A similar second-order condition was obtained earlier by the author for a strong minimum in a much more general problem. In the present paper, we would like to take a narrower perspective than before and thus provide shorter and simpler proofs. In addition, the paper uses the first and second order tangents to the set <italic>U</italic>, defined by the inequality <italic>G</italic>(<italic>u</italic>) ≤ 0. The main difficulty of the proof, clearly shown in the paper, refers to the set, where the gradient <italic>H</italic><italic>u</italic> of the Hamiltonian is small, but the condition of quadratic growth of the Hamiltonian is satisfied. The paper can be valuable for self-explanation and provides a basis for extensions.
</abstract>

A second-order sufficient condition for a weak local minimum in an optimal control problem with an inequality control constraint

<abstract>
<title style='display:none'>Abstract</title>
The continuous auditing technology assures integrity of accounting systems and consequently improves the decision-making process of the small and medium-sized enterprises (SMEs) that implement it. Considering that SMEs located in developing countries function within a more risk prone environment and do not have resources to implement all layers of customized corporate functions in information systems, one argues for their reliance on the features of low complexity of enterprise resource planning (ERP) software to benefit from continuous auditing (CA). The purpose of this study is to relate the understanding of the CA demands and low complexity ERP systems’ technical functionalities in SMEs. Thus, to fulfill this objective, a conceptual model has been drawn to integrate the key concepts related to CA. Four pillars are the core of this model, namely: segregation of duties (SoD) with role-based access control centered on process-based approach (PBA); internal checkpoints; audit trails; and the level of integration of the continuous auditing software. This model was validated through the benchmarking of the implementation of the pillars in three cases of low complexity ERP systems adopted by SMEs in a developing country. The benchmarking/results of the study show significant differences between operational mechanisms of the three ERP software. Namely, the role-based access control exists in the two of the ERP_LC but not in the Brazilian one. Also, there is no check-point in the Brazilian ERP_LC and it does not integrate with continuous audit features. This study distinguishes between the low complexity ERP’s functionalities and the features of a more complex environment, thus bringing an important contribution to the study of low complexity ERP’s readiness for continuous monitoring in SME’s internal auditing processes.
</abstract>

Readiness of Low Complexity ERP for Continuous Auditing in SMEs: The Brazilian Case Study

<abstract>
<title style='display:none'>Abstract</title>
This paper deals with a robust multiobjective optimization problem involving nonsmooth/nonconvex real-valued functions. Under an appropriate constraint qualification, we establish necessary optimality conditions for weakly robust efficient solutions of the considered problem. These optimality conditions are presented in terms of Karush-Kuhn-Tucker multipliers and convexificators of the related functions. Examples illustrating our findings are also given.
</abstract>

Necessary Optimality Conditions for Robust Nonsmooth Multiobjective Optimization Problems

<abstract>
<title style='display:none'>Abstract</title>
In this work, we consider the partial observability problem for finite dimensional dynamical linear systems that are not necessarily observable. For that purpose we introduce the so called “observable subspaces” and “partial observability” to find a way to reconstruct the observable part of the system state. Some characterizations of “observable subspaces” have been provided. The reconstruction of the orthogonal projection of the state on the observable subspace is obtained. We give some examples to illustrate our theoretical approach.
</abstract>

Partial observability of finite dimensional linear systems

<abstract>
<title style='display:none'>Abstract</title>
Optimality conditions for optimal control problems arising in network modeling are discussed. We confine ourselves to the steady state network models. Therefore, we consider only control systems described by ordinary differential equations. First, we derive optimality conditions for the nonlinear problem for a single beam. These conditions are formulated in terms of the <italic>local</italic> Pontryagin maximum principle and the matrix Riccati equation. Then, the optimality conditions for the control problem for networks posed on an arbitrary planar graph are discussed. This problem has a set of independent variables <italic>x</italic>i varying within their intervals [0, <italic>l</italic>i], associated with the corresponding beams at network edges. The lengths <italic>l</italic>i of intervals are not specified and must be determined. So, the optimization problem is non-standard, it is a combination of control and design of networks. However, using a linear change of the independent variables, it can be reduced to a standard one, and we show this. Two simple numerical examples for the single-beam problem are considered.
</abstract>

Network optimality conditions

<abstract>
<title style='display:none'>Abstract</title>
The research, presented in this paper, concernes the controllability of a multi-agent network with a directed, unweighted, cooperative, and time-invariant communication topology. The network’s agents follow linear and heterogeneous dynamics, encompassing first-order, second-order, and third-order differential equations over continuous time. Two classes of neighbour-based linear distributed control protocols are considered: the first one utilises average feedback from relative velocities/relative accelerations, and the second one utilises feedback from absolute velocities/absolute accelerations. Under both protocols, the network’s agents achieve consensus in their states asymptotically. We observe that both of the considered dynamical rules exploit the random-walk normalised Laplacian matrix of the network’s graph. By categorising the agents of the network into leaders and followers, with leaders serving as exogenous control inputs, we analyse the controllability of followers within their state space through the influence of leaders. Specifically, matrix-rank conditions are established to evaluate the leader–follower controllability of the network under both control protocols. These matrix-rank conditions are further refined in terms of the system matrices’ eigenvalues and eigenvectors. The inference diagrams presented in this work provide deeper insights into how leader–follower interactions impact the network controllability. The efficacy of the theoretical findings is validated through numerical examples.
</abstract>

Controllability of consensus heterogeneous multi-agent networks over continuous time scale

<abstract>
<title style='display:none'>Abstract</title>
In this paper the explicit necessary and sufficient conditions for the existence of Luenberger reduced order observer are established. In particular, it is proven that for the given linear time-invariant system of order <italic>n</italic>, having <italic>p</italic> linearly independent outputs and <italic>m</italic> inputs, a Luenberger observer of order (<italic>n</italic> − <italic>p</italic>) can be constructed if and only if the given system is detectable. Furthermore, a procedure is given for the construction of the observer. Our approach is based on the properties of real and polynomial matrices.
</abstract>

On the existence of Luenberger reduced order observer

From the Editors: Some current aspects of publishing of Control and Cybernetics

<abstract>
<title style='display:none'>Abstract</title>
Abstract controlled evolution inclusions are revisited in the Banach spaces setting. The existence of solution is established for each selected control. Then, the input–output (or, control-states) multimap is examined and the Lipschitz continuous well posedness is derived. The optimal control of such inclusions handled in terms of a Bolza problem is investigated by means of the so-called <italic>P</italic>ℱ format of optimization. A strong duality is provided, the existence of an optimal pair is given and the system of optimalty is derived. A Fenchel duality is built and applied to optimal control of convex process of evolution. Finally, it will be shown how the general theory we provided can be applied to a wide class of controled integrodifferental inclusions.
</abstract>

Control and optimization of abstract continuous time evolution inclusions

<abstract>
<title style='display:none'>Abstract</title>
This paper is devoted to the discussion of the exponential stability of a networked hyperbolic system with a circle. Our analysis extends an example by Bastin and Coron about the limits of boundary stabilizability of hyperbolic systems to the case of a networked system that is defined on a graph which contains a cycle. By spectral analysis, we prove that the system is stabilizable while the length of the arcs is sufficiently small. However, if the length of the arcs is too large, the system is not stabilizable. Our results are robust with respect to small perturbations of the arc lengths. Complementing our analysis, we provide numerical simulations that illustrate our findings.
</abstract>

Limits of stabilization of a networked hyperbolic system with a circle

<abstract>
<title style='display:none'>Abstract</title>
Deformations of the computational mesh, arising from optimization routines, usually lead to decrease of mesh quality or even destruction of the mesh. We propose a theoretical framework using pre-shapes to generalize the classical shape optimization and calculus. We define pre-shape derivatives and derive corresponding structure and calculus theorems. In particular, tangential directions are featured in pre-shape derivatives, in contrast to classical shape derivatives, featuring only normal directions. Techniques from classical shape optimization and calculus are shown to carry over to this framework. An optimization problem class for mesh quality is introduced, which is solvable with the use of pre-shape derivatives. This class allows for simultaneous optimization of the classical shape objectives and mesh quality without deteriorating the classical shape optimization solution. The new techniques are implemented and numerically tested for 2D and 3D.
</abstract>

Pre-shape calculus and its application to mesh quality optimization

<abstract>
<title style='display:none'>Abstract</title>
In this article, we study the time-domain decomposition of optimal control problems for systems of semilinear hyperbolic equations and provide an in-depth well-posedness analysis. This is a continuation of our work, Krug et al. (2021) in that we now consider <italic>mixed</italic> two-point boundary value problems. The more general boundary conditions significantly enlarge the scope of applications, e.g., to hyperbolic problems on metric graphs with cycles. We design an iterative method based on the optimality systems that can be interpreted as a decomposition method for the original optimal control problem into <italic>virtual control</italic> problems on smaller time domains.
</abstract>

Time-domain decomposition for optimal control problems governed by semilinear hyperbolic systems with mixed two-point boundary conditions

<abstract>
<title style='display:none'>Abstract</title>
Computational meshes arising from shape optimization routines commonly suffer from decrease of mesh quality or even destruction of the mesh. In this work, we provide an approach to regularize general shape optimization problems to increase both shape and volume mesh quality. For this, we employ pre-shape calculus as established in Luft and Schulz (2021). Existence of regularized solutions is guaranteed. Further, consistency of modified pre-shape gradient systems is established. We present pre-shape gradient system modifications, which permit simultaneous shape optimization with mesh quality improvement. Optimal shapes to the original problem are left invariant under regularization. The computational burden of our approach is limited, since additional solution of possibly larger (non-)linear systems for regularized shape gradients is not necessary. We implement and compare pre-shape gradient regularization approaches for a 2D problem, which is prone to mesh degeneration. As our approach does not depend on the choice of metrics representing shape gradients, we employ and compare several different metrics.
</abstract>

Simultaneous shape and mesh quality optimization using pre-shape calculus

<abstract>
<title style='display:none'>Abstract</title>
We study distributed optimal control problems, governed by space-time fractional parabolic equations (STFPEs) involving time-fractional Caputo derivatives and spatial fractional derivatives of Sturm-Liouville type. We first prove existence and uniqueness of solutions of STFPEs on an open bounded interval and study their regularity. Then we show existence and uniqueness of solutions to a quadratic distributed optimal control problem. We derive an adjoint problem using the right-Caputo derivative in time and provide optimality conditions for the control problem. Moreover, we propose a finite difference scheme to find the approximate solution of the considered optimal control problem. In the proposed scheme, the well-known L1 method has been used to approximate the time-fractional Caputo derivative, while the spatial derivative is approximated using the Grünwald-Letnikov formula. Finally, we demonstrate the accuracy and the performance of the proposed difference scheme via examples.
</abstract>

Distributed optimal control problems driven by space-time fractional parabolic equations

<abstract>
<title style='display:none'>Abstract</title>
A vast majority of research has been performed in the field of hesitant fuzzy sets (HFSs), involving the introduction of some properties, operations, relations and modifications of such sets or considering the application of HFSs in MCDM (multicriteria decision making). On the other hand, no research has been performed in the field of fully hesitant fuzzy equations. Therefore, in this paper, fully hesitant fuzzy equations and dual hesitant fuzzy equations are introduced. First, a method is proposed to solve one-element hesitant fuzzy equations. Then, the proposed method is extended to solve <italic>n</italic>-element hesitant fuzzy equations effectively. Moreover, to show the applicability of the proposed method, it is used to solve a real world problem. Thus, the proposed method is applied to determine market equilibrium price. Also, some other numerical examples are presented to better show the performance of the proposed method.
</abstract>

Introducing hesitant fuzzy equations and determining market equilibrium price

<abstract>
<title style='display:none'>Abstract</title>
This paper presents a practical case involving a shopping mall location problem in the northeast countryside of Brazil. In this problem, conflicting objectives have been expressed in terms of seven criteria. Then, ten cities of the northeastern countryside have been selected to compose the space of actions. The problem plays a special role since Brazil is a big country that requires investments in the countryside. Thus, the shopping mall aims to stimulate economic growth in the respective region. In the study, this multi-objective problem is solved using the FITradeoff method. In FITradeoff, the combination of the paradigms of holistic evaluation and elicitation by decomposition in preference modeling are well explored, bringing different perspectives for the decision-maker during the decision process.
</abstract>

Using the FITradeoff method to solve a shopping mall location problem in the northeastern countryside of Brazil

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Volume 50 (2021): Issue 1 (March 2021) Special Issue Dedicated to the memory of Gregory E. Kersten Guest Edited by Tomasz Wachowicz, Ewa Roszkowska and Bogumił Kamiński