Abstract. Control problems for the 3D model of acoustic scattering which describes scattering acoustic waves by a permeable obstacle with the form of a spherical layer are considered. These problems arise while developing the design technologies of acoustic cloaking devices using the wave flow method. The solvability of direct and control problems for the acoustic scattering model under study is proved. The sufficient conditions which provide local uniqueness and stability of optimal solutions are established.
IntroductionRecent significant research has focused on design of invisibility cloaking devices for material bodies. Beginning with pioneering papers [1,2,3], a large number of publications was devoted to developing different methods of solving the cloaking problems. The transformation optics (TO) method proposed in [2] is the most popular cloaking method lying on the base of the so-called passive cloaking strategy. The first applications of the TO method were connected with solving electromagnetic cloaking problems. Then after proof of the equivalence of the 2D Maxwell equations to the 2D equations of acoustics of anisotropic medium, this theory was expanded to acoustic cloaking at first in 2D case [4] and then in general 3D case [5,6]. We also mention the papers [7,8,9] devoted to the proof of existence of acoustic cloaking shells and to the studies of properties of solutions of the acoustic cloaking problem. In [10], the invisibility problem in X-ray tomography was studied.It should be emphasized that the technical realization of solutions obtained in cited references is connected with substantial difficulties. One of approaches of overcoming these difficulties consists of replacing the exact cloaking problem by the approximate cloaking problem for which solutions admit relatively simple technical realization (see, e.g., [11,12,13,14] and references therein). Another approach is based on using the optimization method of solving inverse problems. This method was applied in papers [15,16] devoted to numerical analysis of 2D cloaking problems and in [17,18,19] when studying impedance cloaking problems.Optimization method is also applied in this paper while theoretical analysis of acoustic cloaking problems in which the cloaking effect is achieved due to the choice of variable parameters of inhomogeneous isotropic liquid medium filling the acoustic cloaking shell. We begin with formulation of the direct scattering problem. Let Ω be the bounded domain in R 3 which is topologically equivalent to the spherical layer with the interior domain Ω i , the external domain Ω ∞ e and, in the general case, with the curvilinear boundary Γ consisting of two components:
We consider the optimization problem for the 2D model of magnetic scattering by a permeable obstacle having the form of a circular ring. Problems of this type arise while developing the design technologies of magnetic cloaking devices using the optimization method. The solvability of direct and control problems for the magnetic scattering model under study is proved. The sufficient conditions which provide local uniqueness and stability of optimal solutions are established.
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