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Main publications

  1. Kogut, P.I. and Leugering, G., Asymptotic Analysis of State Constrained Semilinear Optimal Control Problems. Journal of Opt. Theory and Appl. (JOTA), Vol.135, No.2, 2007, pp. 301–321. DOI: 10.1007/s10957-007-9282-1

The asymptotic behavior of state-constrained semilinear optimal control problems for distributed-parameter systems with variable compact control zones is investigated. We derive conditions under which the limiting problems can be made explicit.

  1. Dovzhenko, A.V., Kogut, P.I., and Manzo, R., On the concept of Г-convergence for locally compact vector-valued mappings. OFar East Journal of Applied Mathematics, Vol.60, Issue 1, 2011, p.1-39.

This paper deals with a new concept of limit for sequences of locally compact vector-valued mappings in normed spaces. We generalize the well-known concept of Γ-convergence to the so-called ΓΛ,μ – convergence in vector-valued case. To this aim, we study the link between the lower semicontinuity property of vector-valued mappings and the topological properties of their coepigraphs. We show that, if the objective space is partially ordered by a pointed cone with nonempty interior, then coepigraphs are stable with respect to their closure and, moreover, the locally semicompact vector-valued mappings with closed coepigraphs are lower continuous. Using these results, we establish the relationship between ΓΛ,μ -convergence of the sequences of mappings and K-convergence of their coepigraphs in the sense of Kuratowski and study the main topological properties of ΓΛ,μ -limits.

  1. Kogut, P.I. and Leugering, G., Homogenization of optimal control problems for one-dimensional elliptic equations on periodic graph. ESAIM: Control, Optimization and Calculus of Variations (COCV), Vol. 15, 2009, pp. 471–498. DOI: 10.1007/s10957-007-9282-1

We are concerned with the asymptotic analysis of optimal control problems for 1-D partial differential equations defined on a periodic planar graph, as the period of the graph tends to zero. We focus on optimal control problems for elliptic equations with distributed and boundary controls. Using approaches of the theory of homogenization we show that the original problem on the periodic graph tends to a standard linear quadratic optimal control problem for a two-dimensional homogenized system, and its solution can be used as suboptimal controls for the original problem.

  1. D’Apice, C., De Maio, U., Kogut, P.I. and Manzo R., On the Solvability of an Optimal Control Problem in Coefficients for Ill-Posed Elliptic Boundary Value Problems. Electronic Journal of Differential Equations, Vol. 2014, No. 166, 2014, 1-23.

We study an optimal control problem (OCP) associated to a linear elliptic equation − div(A(x)∇y + C(x)∇y) = f. The characteristic feature of this control object is the fact that the matrix C(x) is skew-symmetric and belongs to L2 -space (rather than L). We adopt a symmetric positive de- fined matrix A(x) as control in L(Ω; RN×N). In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions, we prove that the corresponding OCP, under rather general assumptions on the class of admissible controls, is well-posed and admits a nonempty set of solutions. The main trick we apply to the proof of the existence result is the approximation of the original OCP by regularized OCPs in perforated domains with fictitious boundary controls on the holes.

  1. D’Apice, C., De Maio, U. and Kogut, P.I., Suboptimal boundary controls for elliptic equation in critically perforated domain. Annales de I’Institut Henri Poincare (C), Nonlinear Analysis, Vol.25, Issue 6, 2008, pp. 1073–1101. DOI: 10.1016/j.anihpc.2007.07.001

In this paper we study the asymptotic behaviour, as ε tends to zero, of a class of boundary optimal control problems Pε, set in ε-periodically perforated domain. The holes have a critical size with respect to ε-sized mesh of periodicity. The support of controls is contained in the set of boundaries of the holes. This set is divided into two parts, on one part the controls are of Dirichlet type; on the other one the controls are of Neumann type. We show that the optimal controls of the homogenized problem can be used as suboptimal ones for the problems Pε.

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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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