# Main publications

**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*

**Abstract**

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.

**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.*

**Abstract**

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 Γ^{Λ,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 Γ^{Λ,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
Γ^{Λ,μ} -limits.

**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*

**Abstract**

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.

**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.*

**Abstract**

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 *L*^{2}
-space (rather than *L*^{∞}). We adopt a symmetric positive de-
fined matrix A(x) as control in *L*^{∞}(Ω; R^{N×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.

**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*

**Abstract**

In this paper we study the asymptotic behaviour, as ε tends to zero, of a class of boundary optimal control problems P_{ε}, 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 P_{ε}.

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