In this article, we consider the optimal control problem governed by the wave equation in a 2-dimensional domain
Ω
ϵ
in which the state equation and the cost functional involves highly oscillating ...periodic coefficients
A
ϵ
and
B
ϵ
, respectively. This paper aims to examine the limiting behavior of optimal control and state and identify the limit optimal control problem, which involves the influences of the oscillating coefficients.
Abstract
We treat an inverse electrical conductivity problem which deals with the reconstruction of nonlinear electrical conductivity starting from boundary measurements in steady currents ...operations. In this framework, a key role is played by the Monotonicity Principle, which establishes a monotonic relation connecting the unknown material property to the (measured) Dirichlet-to-Neumann operator (DtN). Monotonicity Principles are the foundation for a class of non-iterative and real-time imaging methods and algorithms. In this article, we prove that the monotonicity principle for the Dirichlet Energy in nonlinear problems holds under mild assumptions. Then, we show that apart from linear and
p
-Laplacian cases, it is impossible to transfer this monotonicity result from the Dirichlet Energy to the DtN operator. To overcome this issue, we introduce a new boundary operator, identified as an average DtN operator.
In this paper, an equilibrium problem for 2D non-homogeneous anisotropic elastic body is considered. It is assumed that the body has a thin elastic inclusion and a thin rigid inclusion. A connection ...between the inclusions at a given point is characterized by a junction stiffness parameter. The elastic inclusion is delaminated, thus forming an interfacial crack with the matrix. Inequality-type boundary conditions are imposed at the crack faces to prevent interpenetration. Existence of solutions is proved; different equivalent formulations of the problem are discussed; junction conditions at the connection point are found. A convergence of solutions as the junction stiffness parameter tends to zero and to infinity as well as the rigidity parameter of the elastic inclusion tends to infinity is investigated. An analysis of limit models is provided. An optimal control problem is analyzed with the cost functional equal to the derivative of the energy functional with respect to the crack length. A solution existence of an inverse problem for finding the junction stiffness and rigidity parameters is proved.
Abstract This paper deals with the Monotonicity Principle (MP) for nonlinear materials with piecewise growth exponent. The results obtained are relevant because they enable the use of a fast imaging ...method based on MP, applied to a wide class of problems with two or more materials, at least one of which is nonlinear. The treatment is very general and makes it possible to model a wide range of practical configurations such as superconducting (SC), perfect electrical conducting (PEC) or perfect electrical insulating (PEI) materials. A key role is played by the average Dirichlet-to-Neumann operator, introduced in Corbo Esposito et al (2021 Inverse Problems 37 045012), where the MP for a single type of nonlinearity was treated. Realistic numerical examples confirm the theoretical findings.
In a planar infinite strip with a fast oscillating boundary we consider an elliptic operator assuming that both the period and the amplitude of the oscillations are small. On the oscillating boundary ...we impose Dirichlet, Neumann or Robin boundary condition. In all cases we describe the homogenized operator, establish the uniform resolvent convergence of the perturbed resolvent to the homogenized one, and prove the estimates for the rate of convergence. These results are obtained as the order of the amplitude of the oscillations is less, equal or greater than that of the period. It is shown that under the homogenization the type of the boundary condition can change.
We study the optimal control problem of a second order linear evolution equation defined in two-component composites with
ε
-periodic disconnected inclusions of size
ε
in presence of a jump of the ...solution on the interface that varies according to a parameter
γ
. In particular here the case
γ
<
1
is analyzed. The optimal control theory, introduced by Lions (Optimal Control of System Governed by Partial Differential Equations, 1971), leads us to characterize the control as the solution of a set of equations, called optimality conditions. The main result of this paper proves that the optimal control of the
ε
-problem, which is the unique minimum point of a quadratic cost functional
J
ε
, converges to the optimal control of the homogenized problem with respect to a suitable limit cost functional
J
∞
. The main difficulties are to find the appropriate limit functional for the control of the homogenized system and to identify the limit of the controls.
In this paper we study the asymptotic behaviour of an exact controllability problem for a second order linear evolution equation defined in a two-component composite with ε-periodic disconnected ...inclusions of size ε. On the interface we prescribe a jump of the solution that varies according to a real parameter γ. In particular, we suppose that −1<γ≤1. The case γ=1 is the most interesting and delicate one, since the homogenized problem is represented by a coupled system of a P.D.E. and an O.D.E., giving rise to a memory effect. Our approach to exact controllability consists in applying the Hilbert Uniqueness Method, introduced by J.-L. Lions, which leads us to the construction of the exact control as the solution of a transposed problem. Our main result proves that the exact control and the corresponding solution of the ε-problem converge to the exact control of the homogenized problem and to the corresponding solution respectively.
Dans cet article nous étudions le comportement asymptotique d'un problème de contrôlabilité exacte pour une équation d'évolution linéaire du second ordre, dans un milieu composite à deux composantes présentant des inclusions ε-périodiques de taille ε. Sur l'interface entre les deux composantes on prescrit un saut de la solution qui est proportionnel par un facteur εγ à la dérivée conormale. On suppose −1<γ≤1. Le cas γ=1, plus délicat, est aussi le plus intéressant, puisque le problème homogénéisé est un système couplé de deux équations, une EDP et une EDO, ce qui génère un effet de mémoire. Notre approche à la contrôlabilité exacte consiste à appliquer la méthode HUM (Hilbert Uniqueness Method), introduite par J.-L. Lions, ce qui nous conduit à la construction du contrôle exact comme solution d'un problème transposé. Notre résultat principal montre que le contrôle exact et la solution correspondante du ε-problème convergent respectivement vers le contrôle exact du problème homogénéisé et la solution correspondante.
Junction of quasi-stationary ferromagnetic wires Chacouche, Khaled; Faella, Luisa; Perugia, Carmen
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni,
01/2020, Letnik:
31, Številka:
1
Journal Article
Recenzirano
In this paper we study the asymptotic behavior of the solutions of time dependent micromagnetism problem in a structure consisting of two joined thin wires. We assume that the volumes of the two ...parts composing the multi-structure vanish with same rate. We obtain two 1$D$ limit problems coupled by a junction condition on the magnetization. The limit problem remains non-convex, but now it becomes completely local.
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