DCDS
On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients
Peter I. Kogut
Discrete & Continuous Dynamical Systems - A 2014, 34(5): 2105-2133 doi: 10.3934/dcds.2014.34.2105
We study an optimal boundary control problem (OCP) associated to a linear elliptic equation $-\mathrm{div}\,\left(\nabla y+A(x)\nabla y\right)=f$. The characteristic feature of this equation is the fact that the matrix $A(x)=[a_{ij}(x)]_{i,j=1,\dots,N}$ is skew-symmetric, $a_{ij}(x)=-a_{ji}(x)$, measurable, and belongs to $L^2$-space (rather than $L^\infty$). 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.
keywords: variational convergence fictitious control. non-variational solutions boundary control Diffusion equations
MCRF
Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result
Thierry Horsin Peter I. Kogut
Mathematical Control & Related Fields 2015, 5(1): 73-96 doi: 10.3934/mcrf.2015.5.73
In this paper we study an optimal control problem (OCP) associated to a linear elliptic equation on a bounded domain $\Omega$. The matrix-valued coefficients $A$ of such systems is our control in $\Omega$ and will be taken in $L^2(\Omega;\mathbb{R}^{N\times N})$ which in particular may comprises the case of unboundedness. Concerning the boundary value problems associated to the equations of this type, one may exhibit non-uniqueness of weak solutions--- namely, approximable solutions as well as another type of weak solutions that can not be obtained through the $L^\infty$-approximation of matrix $A$. Following the direct method in the calculus of variations, we show that the given OCP is well-possed and 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 not be attainable through the $L^\infty$-approximation of the original problem.
keywords: variational convergence variational solutions existence result. Control in coefficients
MCRF
Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions
Thierry Horsin Peter I. Kogut Olivier Wilk
Mathematical Control & Related Fields 2016, 6(4): 595-628 doi: 10.3934/mcrf.2016017
In this paper we study we study a Dirichlet optimal control problem associated with a linear elliptic equation the coefficients of which we take as controls in the class of integrable functions. The characteristic feature of this control object is the fact that the skew-symmetric part of matrix-valued control $A(x)$ belongs to $L^2$-space (rather than $L^\infty)$. In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions, the corresponding OCP, under rather general assumptions on the class of admissible controls, is well-posed and admits a nonempty set of solutions [9]. However, the optimal solutions to such problem may have a singular character. We show that some of optimal solutions can be attainable by solutions of special optimal control problems in perforated domains with fictitious boundary controls on the holes.
keywords: non-variational solutions Control in coefficients fictitious control. variational convergence
DCDS-B
Boundary velocity suboptimal control of incompressible flow in cylindrically perforated domain
Ciro D’Apice Umberto De Maio Peter I. Kogut
Discrete & Continuous Dynamical Systems - B 2009, 11(2): 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 $\Omega_{\epsilon}$. 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.
keywords: optimal control variational convergence vorticity minimization Dirichlet control Navier-Stokes equation
NHM
On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains
Ciro D'Apice Peter I. Kogut Rosanna Manzo
Networks & Heterogeneous Media 2014, 9(3): 501-518 doi: 10.3934/nhm.2014.9.501
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.
keywords: entropy solutions optimal control relaxation supply chains Conservation laws Henig dilating cone.
NHM
On optimization of a highly re-entrant production system
Ciro D'Apice Peter I. Kogut Rosanna Manzo
Networks & Heterogeneous Media 2016, 11(3): 415-445 doi: 10.3934/nhm.2016003
We discuss the optimal control problem stated as the minimization in the $L^2$-sense of the mismatch between the actual out-flux and a demand forecast for a hyperbolic conservation law that models a highly re-entrant production system. The output of the factory is described as a function of the work in progress and the position of the switch dispatch point (SDP) where we separate the beginning of the factory employing a push policy from the end of the factory, which uses a quasi-pull policy. The main question we discuss in this paper is about the optimal choice of the input in-flux, push and quasi-pull constituents, and the position of SDP.
keywords: Conservation laws optimal control re-entrant systems. existence result

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