Preface

The Summer Workshop on Interval Methods (SWIM) is an annual meeting initiated in 2008 by the French MEA working group on Set Computation and Interval Techniques of the French research group on Automatic Control. A special focus of the MEA group is on promoting interval analysis techniques and applications to a broader community of researchers, facilitated by such multidisciplinary workshops. Since 2008, SWIM has become a keystone event for researchers dealing with various aspects of interval and set-based methods. In 2019, the 12th edition in this workshop series was held at ENSTA Paris, France, with a total of 25 talks. Traditionally, workshops in the series of SWIM provide a platform for both theoretical and applied researchers who work on the development, implementation, and application of interval methods, verified numerics, and other related (set-membership) techniques.For this edition, given talks were in the fields of the verified solution of initial value problems for ordinary differential equations, differential-algebraic system models, and partial differential equations, scientific computing with guaranteed error bounds, the design of robust and fault-tolerant control systems, the implementation of corresponding software libraries, and the usage of the mentioned approaches for a large variety of system models in areas such as control engineering, data analysis, signal and image processing. Seven papers were selected for submission to this Acta Cybernetica special issue. After a two turn peer-review process, six high-quality articles were selected for publication in this special issue. Three papers propose a contribution regarding differential equations, two papers focus on robust control, and one paper considers fault detection.

Linear quadratic optimization for fractional order differential algebraic system of Riemann-Liouville type

In this article, the linear quadratic optimization problem subject to fractional order differential algebraic systems of Riemann-Liouville type is studied. The goal of this article is to find the optimal control-state pairs satisfying the dynamic constraint of the form a fractional order differential algebraic systems such that the linear quadratic objective functional is minimized. The transformation method is used to find the optimal controlstate pairs for this problem. The optimal control-state pairs is stated in terms of Mittag-Leffler function.

Nonlinear vibration of a lumped system with springs-in-series

The paper deals with the dynamics of a lumped mass mechanical system containing two nonlinear springs connected in series. The external harmonic excitation, linear and nonlinear damping are included into considerations. The mathematical model contains both differential and algebraic equations, so it belongs to the class of dynamical systems governed by the differential–algebraic system of equations (DAEs). An approximate analytical approach is used to solve the initial value problem for the DAEs. We adopt the multiple scales method (MSM) that allows one to obtain the sufficiently correct approximate solutions both far from the resonance and at the resonance conditions. The steady and non-steady resonant vibrations are analyzed by employing the modulation equations of the amplitudes and phases which are yielded by the MSM procedure.

Stability analysis of fractional-order linear neutral delay differential–algebraic system described by the Caputo–Fabrizio derivative

This paper is concerned with the asymptotic stability of linear fractional-order neutral delay differential–algebraic systems described by the Caputo–Fabrizio (CF) fractional derivative. A novel characteristic equation is derived using the Laplace transform. Based on an algebraic approach, stability criteria are established. The effect of the index on such criteria is analyzed to ensure the asymptotic stability of the system. It is shown that asymptotic stability is ensured for the index-1 problems provided that a stability criterion holds for any delay parameter. Also, asymptotic stability is still valid for higher-index problems under the conditions that the system matrices have common eigenvectors and each pair of such matrices is simultaneously triangularizable so that a stability criterion holds for any delay parameter. An example is provided to demonstrate the effectiveness and applicability of the theoretical results.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

Matrix boundary-value problems for differential equations with p-Laplacian

We consider the boundary-value problem for a linear system of differential equations with matrix p-Laplacian, which is reduced to the traditional differential-algebraic system with an unknown in the form of the vector function. A generalization of various boundary-value problems for differential equations with p-Laplacian, which preserves the features of the solution of such problems, namely, the lack of uniqueness of the solution and, in this case, the dependence of the desired solution on an arbitrary function, is given.

Application of integral method for investigating the boriding kinetics of AISI 316 steel

The present work is dealing with the modelling of boriding kinetics of AISI 316 steel in the temperature range 1123–1273 K. A diffusion model based on the integral method was used in order to investigate the kinetics of formation of FeB and Fe2B layers and that of diffusion zone formed on AISI 316 steel by considering the presence of boride incubation times. By using a particular solution of the resulting differential algebraic system, the diffusion coefficients in FeB, Fe2B and diffusion zone (DZ) were estimated as well as the corresponding values of activation energies. Finally, this present diffusion model has been experimentally validated for two additional boriding conditions (1243 K for 3 and 5 h of treatment). A good concordance was observed between the experimental and the simulated results in terms of layers’ thicknesses.

About the equilibrium positions of a matrix differential-algebraic boundary value problem

In the article we found the solvability conditions and the construction of the generalized Green operator of the linear Noetherian matrix differential-algebraic boundary value problem. We obtained sufficient conditions of transformationsof the matrix differential-algebraic equation to a traditional differential-algebraic equation with an unknown in the form of a column vector. The problem that reviewed in the article continues the study of solvability conditions for the linear Noetherian boundary value problems given in the monographs of M.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina, A.M. Samoilenko and A.A. Boichuk. We investigated the general case when the linear bounded operator corresponding to the homogeneous part of the linear Cauchy problem for the matrix differential-algebraic system does not have the reverse operator. We introduced the definition of the equilibrium positions of the matrix differential-algebraic system and the matrix differential-algebraic boundary-value problem to solve the matrix differential-algebraic boundary-value problem. We proposed sufficient conditions of existence and constructive schemes for finding the equilibrium positions of the matrix differential-algebraic system and the matrix differential-algebraic boundary value problem. The cases~of equilibrium positions of the matrix differential-algebraic system, which are constant matrices, and equilibrium positions depending on an independent variable are considered separately. To solve the matrix differential-algebraic boundary-value problem, we used the original solvability conditions and~the construction of the general solution of the Sylvester-type matrix equation, while the Moore-Penrose matrix pseudoinverse technique was essentially used. In the article we constructed the generalized Green operator of the linear Noetherian matrix differential-algebraic boundary value problem. The proposed solvability conditions and the construction of the generalized Green operator of the linear Noetherian matrix differential-algebraic boundary value problem, were illustrated in detail with examples.

On the solution of a linear Noetherian boundary-value problem for a differential-algebraic system with lumped delay by the method of least squares

The conditions of existence and the construction of pseudosolutions, being the best by the method of least squares, of a Noetherian differential-algebraic boundary-value problem with concentrated delay, respectively, are determined.