Optimal Control on Perforated Domains

Kesavan, S.; Paulin, J. Saint Jean

doi:10.1006/jmaa.1998.6185

Cited by 38 publications

(31 citation statements)

References 2 publications

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“…As is well known, the practical significance of the solution of the problem of homogenization of such objects follows from the needs of mechanics of composite materials and the theory of filtration in porous media. In particular, as is shown in works on problems of limit analysis of boundary-value problems in perforated domains (see, for example, [4,[11][12][13][14][15][16][17]), their solutions converge to the solution of a boundary-value problem whose associated operator is the sum of a G-limit operator and some additional lowest term that characterizes the perforation structure of the initial domain.…”

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confidence: 94%

G*-Convergence of Nonlinear Operators in the Theory of Homogenization of Control Objects

Kogut

2005

Cybern Syst Anal

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517.9To study problems of homogenization of one class of control objects described by nonlinear operator equations, the concept of G * -convergence of nonlinear operators is introduced. The sufficient conditions of identifiability of homogenized objects are obtained.Keywords: homogenization, elliptic operator, convergence in the sense of Kuratowski, control object, G-limit, operator graph.The objective of this work is the study of limit behavior of solutions of nonlinear operator equations of the formand problems of homogenization of control objects connected with these equations. To date, the problem of homogenization of differential partial differential equations and operator equations in Banach spaces is considered in many publications (see, for example, [1-10]). However, a distinctive feature of the majority of the results obtained is that, as a rule, a sequence of elements { } f e is assumed to be convergent in the strong topology of the space Y * . In this case, a homogenized object exists and can be identified in terms of the so-called G-limit operators. At the same time, a basic premise of the present work is that the property of compactness of the sequence { } f e being considered takes place only with respect to the weak topology of Y * . In this case, the fact is essential that the sequence { } f e does not contain any strongly convergent subsequence. As is well known, the practical significance of the solution of the problem of homogenization of such objects follows from the needs of mechanics of composite materials and the theory of filtration in porous media. In particular, as is shown in works on problems of limit analysis of boundary-value problems in perforated domains (see, for example, [4, 11-17]), their solutions converge to the solution of a boundary-value problem whose associated operator is the sum of a G-limit operator and some additional lowest term that characterizes the perforation structure of the initial domain.In this connection, we note that the apparatus of G-convergence of elliptic operators becomes unsuitable for the study of limit properties of the sequence of solutions of boundary-value problems (1) under the above-mentioned assumptions on functions { } f e . Therefore, to identify the mathematical model of the problem whose solution is the weak limit of the sequence of solutions of problems (1), the concept of G * -convergence of nonlinear operators is introduced in this paper that is a natural generalization of the well-known conception of G-limit analysis. At the same time, it is important to note that, for the same sequence of operators { } A e , their G-and G * -limits coincide in the case when the family of functions { } f e is compact in the strong topology of the space Y * . This is explained by the fact that, in the definition of G * -convergence, not only properties of operators { } A e but also structural properties of functions { } f e are used. We introduce the following agreements. Let Y be a separable reflexive Banach space with a norm || || × over the real numbe...

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confidence: 94%

G*-Convergence of Nonlinear Operators in the Theory of Homogenization of Control Objects

Kogut

2005

Cybern Syst Anal

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“…Fabre, Puel, and Zuazua [13] investigated the homogenization method for approximate controllability of the semilinear case with Dirichlet boundary conditions by means of a fixed point technique. Kesavan and Saint Jean Paulin [16] obtained homogenization results in nonperiodic cases in the framework of H-convergence and extended them to optimal control systems governed by elliptic boundary value problems in perforated domains (see [17]). Donato and Nabil [12] presented the homogenization and correctors for an approximate controllability problem of the linear heat equation with rapidly oscillating coefficients in a periodically perforated domain.…”

Section: Introductionmentioning

confidence: 99%

A Multiscale Approach for Optimal Control Problems of Linear Parabolic Equations

Cao¹,

Li²,

Allegretto³

et al. 2012

SIAM J. Control Optim.

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Abstract. This paper discusses multiscale analysis for optimal control problems of linear parabolic equations with rapidly oscillating coefficients that depend on spatial and temporal variables. There are mainly three new results in the present paper. First, we obtain the convergence results with an explicit convergence rate for the multiscale asymptotic expansions of the solution of the optimal control problem in the case without constraints. Second, for a general bounded Lipschitz polygonal domain, the boundary layer solution is defined and the corresponding convergence results are also derived. Finally, an explicit convergence rate ε 1/2 in the presence of constraint is reported.

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“…The most typical procedure of homogenization consists of the following steps: at first, we write down the necessary optimality conditions for the initial problem; next we find the corresponding limiting relations as ε → 0 and interpret them as necessary optimality conditions for some control problem; then, using the limiting necessary optimality conditions, we recover an optimal control problem which is called the homogenized control problem (see e.g. [2,11,15,16,32]). Thus, if we denote by OCP ε , NOC ε , HOCP, HNOC the original optimal control problem on the ε-level, the corresponding necessary optimality conditions on the ε-level, the homogenized optimal control problem and the homogenized necessary optimality system, respectively, then the above mentioned procedure can be represented in the following diagram:…”

Section: Introductionmentioning

confidence: 99%

Homogenization of constrained optimal control problems for one-dimensional elliptic equations on periodic graphs

Kogut¹,

Leugering²

2008

ESAIM: COCV

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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.Mathematics Subject Classification. 35B27, 35J25, 49J20, 93C20.

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Optimal Control on Perforated Domains

Cited by 38 publications

References 2 publications

G*-Convergence of Nonlinear Operators in the Theory of Homogenization of Control Objects

G*-Convergence of Nonlinear Operators in the Theory of Homogenization of Control Objects

A Multiscale Approach for Optimal Control Problems of Linear Parabolic Equations

Homogenization of constrained optimal control problems for one-dimensional elliptic equations on periodic graphs

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