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Main publications

  1. Buttazzo, G. and Kogut, P.I., On quadratic scalarization of vector optimization problems in Banach spaces. Applicable Analysis, Vol. 93, No. 5, 2014, 994-1009. DOI: 10.1080/00036811.2013.809068

We study vector optimization problems in partially ordered Banach spaces and suppose that the objective mapping possesses a weakened property of lower semicontinuity and make no assumptions on the interior of the ordering cone. We discuss the so-called adaptive scalarization of such problems and show that the corresponding scalar non-linear optimization problems can be by-turn approximated by quadratic minimization problems.

  1. Horsin, T. and Kogut, P.I., Optimal L2-Control Problem in Coefficients for a Linear Elliptic Equation. Submitted to Mathematical Control and Related Fields, 2013, 1-60. DOI: arXiv:1306.2513

In this paper we study an optimal control problem (OCP) associated to a linear elliptic equation on a bounded domain Ω. The matrixvalued coefficients A of such systems is our control in Ω and will be taken in L2(Ω; RN×N) which in particular may comprise some cases of unboundedness. Concerning the boundary value problems associated to the equations of this type, one may face non-uniqueness of weak solutions— namely, approximable solutions as well as another type of weak solutions that can not be obtained through the L-approximation of matrix A. Following the direct method in the calculus of variations, we show that the given OCP is well-posed in the sense that it admits at least one solution. At the same time, optimal solutions to such problem may have a singular character in the above sense. In view of this, we indicate two types of optimal solutions to the above problem: the so-called variational and non-variational solutions, and show that some of that optimal solutions can be attainable by solutions of special optimal boundary control problems.

  1. Kogut, P.I., Kogut, O.P. and Leugering, G., Optimal Control in Matrix-Valued Coefficients for Nonlinear Monotone Problems: Optimality Conditions. Part I. Journal of Analysis and its Applications (ZAA), Volume 34, Issue 1, 2015, 85-108. DOI: 10.4171/ZAA/

In this article we study an optimal control problem for a nonlinear monotone Dirichlet problem where the controls are taken as matrix-valued coefficients in L(Ω; RN×N). For the exemplary case of a tracking cost functional, we derive first order optimality conditions. This is the first part out of two articles. This first part is concerned with the general case of matrix-valued coefficients under some hypothesis, while the second part focuses on the special class of diagonal matrices.

  1. Kogut, P.I., Kogut, O.P. and Leugering, G., Optimal Control in Matrix-Valued Coefficients for Nonlinear Monotone Problems: Optimality Conditions. Part II. Journal of Analysis and its Applications (ZAA), Volume 34, Issue 2, 2015, 199-219. DOI: DOI: 10.4171/ZAA/

In this paper we study an optimal control problem for a nonlinear monotone Dirichlet problem where the controls are taken as the matrix-valued coefficients in L(Ω; RN×N). Given a suitable cost function, the objective is to provide a substantiation of the first order optimality conditions using the concept of convergence in variable spaces. While in the first part [2] optimality conditions have been derived an analysed in the general case under some assumptions on the quasi-adjoint states, in this second part, we consider diagonal matrices and analyse the corresponding optimality system without such assumptions.

  1. Dovzhenko, A.V., Kogut, P.I., and Manzo, R., Epi and Coepi-Analysis of One Class of Vector-Valued Mappings. Optimization. A Journal of Mathematical Programming and Operations Research, 2014, Vol 63, Issue 4, P. 535-557. DOI: DOI: 10.1080/0233193YY

This paper deals with a new characterization of lower semicontinuity of vector-valued mappings in normed spaces. We study the link between the lower semicontinuity property of vector-valued mappings and the topological properties of their epigraphs and coepigraphs, respectively. We show that if the objective space is partially ordered by a pointed cone with nonempty interior, then coepigraphs are stable with respect to the procedure of their closure and, moreover, the locally semicompact vector-valued mappings with closed coepigraphs are lower semicontinuous. Using these results we propose some regularization schemes for vector-valued functions. In the case when there are no assumptions on the topological interior of the ordering cone, we introduce a new concept of lower semicontinuity for vector-valued mappings, the so-called epi-lower semicontinuity, which is closely related with the closedness of epigraphs of such mappings, and study their main properties. All principal notions and assertions are illustrated by numerous examples.

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Journal of Optimization, Differential Equations and their Applications

The "Journal of Optimization, Differential Equations and their Applications” (shortly JODEA) publishes original research articles related with the recent developments both in theory and applications of ordinary differential equations, partial differential equations, integral equations, functional differential equations, stochastic differential equations, optimal control theory, scalar and vector optimization, and other related topics. The journal will also accept papers from some related fields such as functional analysis, probability theory, stochastic analysis, inverse problems, numerical computation, mathematical finance, game theory, system theory, etc., provided that they have some intrinsic connections with control theory and differential equations. Papers employing differential equations as tools serving the cause of interdisciplinary areas such as physical, biological, environmental and health sciences, mechanics and engineering are encouraged. The goal is to provide a complete and reliable source of mathematical methods and results in these fields.

The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools. To be published in this journal, an original paper must be correct, new, nontrivial and of interesting to readers.

JODEA is issued two times per year. It is edited by a group of international leading experts in mathematical control theory, differential equations and related fields. The journal is essential reading for scientists and researchers who wish to keep abreast of the latest developments in the field of differential equations and their applications. The journal is founded in 2018 by Oles Honchar Dnipro National Universuty for researchers and PhD-students who are working in the field of mathematics and applied mathematics. JODEA is printed according to the decision of the Academic council of the Oles Honchar Dnipro National University and is continuation of the journal "Bulletin of the Dnipropetrovsk University. Series: Modeling" (2009 – 2017, ISSN (Print): 2312-4547, ISSN (Online): 2415-7325) and a series of releases of the collection of scientific works "Differential equations and their applications" which were annually printed during the period from 1968 to 2008.

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