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- Kogut, P.I. and Leugering, G., Optimal L1 -Control in Coefficients for Dirichlet Elliptic Problems: H-Optimal Solutions. Journal for Analysis and its Applications, Vol.31, Issue 1, 31-53. DOI: 10.4171/ZAA/
In this paper we study a Dirichlet optimal control problem associated with a linear elliptic equation the coefficients of which we take as controls in L1 (Ω). In particular, when the coefficient matrix is taken to satisfy the decomposition B(x) = ρ(x)A(x) with a scalar function ρ, we allow the ρ to degenerate. Such problems are related to various applications in mechanics, conductivity and to an approach in topology optimization, the SIMP-method. Since equations of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions, we show that the optimal control problem in the coefficients can be stated in different forms depending on the choice of the class of admissible solutions. Using the direct method in the Calculus of variations, we discuss the solvability of the above optimal control problems in the so-called class of H-admissible solutions.
- D’Apice, C., Kogut, P.I. and Manzo, R., Efficient Controls for Traffic Flow on Networks. Journal of Dynamical and Control Systems, Vol. 16, No. 3, July 2010, 407–437.
We study traffic flow models for road networks in vectorvalued optimization statement, where the flow is controlled at the nodes of the network. We consider the case where an objective mapping possesses a weakened property of upper semicontinuity and make no assumptions on the interior of the ordering cone. We derive suf- ficient conditions for the existence of efficient controls of the traffic problem and discuss the scalarization approach to its solution. We also prove the existence of the so-called generalized efficient controls.
- Kogut, P.I., Manzo, R. and Nechay, I.V., On existence of efficient solutions to vector optimization problems in Banach spaces. Note di Matematica 30 (2010) no. 1, 25–39. DOI: 10.1285/i15900932v30n1p25
In this paper, we present a new characterization of lower semicontinuity of vectorvalued mappings and apply it to the solvability of vector optimization problems in Banach spaces. With this aim we introduce a class of vector-valued mappings that is more wider than the class of vector-valued mappings with the “typical” properties of lower semi-continuity including quasi and order lower semi-continuity. We show that in this case the corresponding vector optimization problems have non-empty sets of efficient solutions.
- D’Apice, C., Kogut, P.I. and Manzo, R., On Approximation of Entropy Solutions for One System of Nonlinear Hyperbolic Conservation Laws with Impulse Source Terms. Journal of Control Science and Engineering, 2010, Vol. 46, No. 2. p.85-119. DOI: 10.1155/2010/982369
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study one class of nonlinear fluid dynamic models with impulse source terms. The model consists of a system of two hyperbolic conservation laws: a nonlinear conservation law for the goods density and a linear evolution equation for the processing rate. We consider the case when influx-rates in the second equation take the form of impulse functions. Using the vanishing viscosity method and the so-called principle of fictitious controls, we show that entropy solutions to the original Cauchy problem can be approximated by optimal solutions of special optimization problems.
- D’Apice, C., De Maio, U. and Kogut, P.I., Boundary Velocity Suboptimal Control of Incompressible Flow in Cylindrically Perforated Domain. Discrete and Continuous Dynamical Systems Series B, Volume 11, Number 2, March 2009 pp. 283–314 . DOI: 10.3934/dcdsb.2009.11.283
In this paper we study an optimal boundary control problem for the 3D steady-state Navier-Stokes equation in a cylindrically perforated domain Ωε. The control is the boundary velocity field supported on the ‘vertical’ sides of thin cylinders. We minimize the vorticity of viscous flow through thick perforated domain. We show that an optimal solution to some limit problem in a non-perforated domain can be used as basis for the construction of suboptimal controls for the original control problem. It is worth noticing that the limit problem may take the form of either a variational calculation problem or an optimal control problem for Brinkman’s law with another cost functional, depending on the cross-size of thin cylinders.
- Buttazzo, G. and Kogut, P.I., Weak optimal controls in coefficients for linear elliptic problems. Revista Matematica Complutense, Vol.24, 2011, pp. 83–94. DOI: 10.1007/s13163-010-0030-y
In this paper we study an optimal control problem associated to a linear degenerate elliptic equation with mixed boundary conditions. The equations of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions. We adopt the weight function as a control in L1(Ω). Using the direct method in the Calculus of variations, we discuss the solvability of this optimal control problem in the class of weak admissible solutions.
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