# Main publications

**Kogut, P.I. and Leugering, G.,**Optimal and approximate boundary controls of an elastic body with quasistatic evolution of damage.*Mathematical Methods in the Applied Sciences, Volume 38, Issue 13, pages 2739–2760, 15 September 2015.**DOI: 10.1002/sim.0000*

**Abstract**

In this paper we study an optimal control problem for the mixed boundary value problem for an elastic body with quasistatic evolution of an internal damage variable. We suppose that the evolution of microscopic cracks and cavities responsible for the damage is described by a nonlinear parabolic equation. A density of surface traction p acting on a part of boundary of an elastic body Ω it taken as a boundary control. Since the initial-boundary value problem of this type can exhibit the Lavrentie phenomenon and non-uniqueness of weak solutions, we deal with the solvability of this problem in the class of weak variational solutions. Using the convergence concept in variable spaces and following the direct method in Calculus of Variations, we prove the existence of optimal and approximate solutions to the optimal control problem under rather general assumptions on the quasistatic evolution of damage.

**Kogut, P.I., Manzo, R. and Nechay, I.V.,**Generalized efficient solutions to one class of vector optimization problems in Banach spaces.*Australian Journal of Mathematical Analysis and Applications, 2010, Vol.7, Issue 1., pp. 1–27.*

**Abstract**

In this paper, we study vector optimization problems in Banach spaces for essentially nonlinear operator equations with additional control and state constraints. We assume that an objective mapping possesses a weakened property of lower semicontinuity and make no assumptions on the interior of the ordering cone. Using the penalization approach we derive both sufficient and necessary conditions for the existence of efficient solutions of the above problems. We also prove the existence of the so-called generalized efficient solutions via the scalarization of some penalized vector optimization problems.

**D’Apice, C., De Maio, U. and Kogut, P.I.,**Gap phenomenon in the homogenization of parabolic optimal control problems.*IMA Journal of Mathematical Control and Information (Cambridge University), Vol.25, 2008, pp. 461–480.**DOI: 10.1093/imamci/dnn010*

**Abstract**

In this paper, we study the asymptotic behaviour of a parabolic optimal control problem in a domain Ω_{ε} ⊂ R^{n}, whose boundary ∂Ω_{ε} contains a highly oscillating part. We consider this problem with two different classes of Dirichlet boundary controls, and, as a result, we provide its asymptotic analysis with respect to the different topologies of homogenization. It is shown that the mathematical descriptions of the homogenized optimal control problems have different forms and these differences appear not only in the state equation and boundary conditions but also in the control constraints and the limit cost functional.

**D’Apice, C., Kogut, P.I. and Manzo, R.,**On Relaxation of State Constrained Optimal Control Problem for a PDE-ODE Model of Supply Chains.*Networks and Heterogeneous Media, Volume 9, Issue 3, 2014, Pages 501-518.**DOI: 10.3934/nhm.2014.9.501*

**Abstract**

We discuss the optimal control problem (OCP) stated as the minimization of the queues and the difference between the effective outflow and a desired one for the continuous model of supply chains, consisting of a PDE for the density of processed parts and an ODE for the queue buffer occupancy. The main goal is to consider this problem with pointwise control and state constraints. Using the so-called Henig delation, we propose the relaxation approach to characterize the solvability and regularity of the original problem by analyzing the corresponding relaxed OCP.

**Kogut, P.I.,**On Approximation of an Optimal Boundary Control Problem for Linear Elliptic Equation with Unbounded Coefficients.*to appear in Discrete and Continuous Dynamical Systems – Series A (DCDS-A), Vol.34, No.5, 2014, 2105-2133.**DOI: doi:10.3934/dcds.2014.34.xx*

**Abstract**

We study an optimal boundary control problem (OCP) associated
to a linear elliptic equation −div (∇y + A(x)∇y) = f. The characteristic feature of this equation is the fact that the matrix A(x) = [a_{ij}(x)]_{i,j=1,…,N}
is skew-symmetric, a_{ij}(x) = −a_{ij}(x), measurable, and belongs to *L*^{2}-space (rather than *L*^{∞}). In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions — namely, they have approximable solutions as well as another type of weak solutions that can not be obtained through an approximation of matrix A, the corresponding OCP is well-possed and admits a unique solution. At the same time, an optimal solution to such problem can inherit a singular character of the original matrix A. We indicate two types of optimal solutions to the above problem: the so-called variational and non-variational solutions, and show that each of that optimal solutions can be attainable by solutions of special optimal boundary control problems.

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