By Olena Bulatsyk, Boris Z. Katsenelenbaum, Yury P. Topolyuk, Nikolai N. Voitovich
This is often the single publication on hand in English language to contemplate inverse and optimization difficulties during which section box distributions are used as optimizing capabilities. The mathematical approach used pertains to nonlinear essential equations, with numerical tools constructed and utilized to concrete difficulties. Written by way of a crew of exceptional and well known specialists within the box, this monograph will entice all these facing the research, layout, and optimization of electromagnetic and acoustic radiating and transmitting units and structures, whereas additionally being of curiosity to mathematicians engaged on the speculation of nonlinear vital equations.
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Extra resources for Phase Optimization Problems: Applications in Wave Field Theory
The function F is assumed to be zero in the complement of the angular domain Ω to the period. 1, respectively. 2. 58) is the outgoing energy part outside the angle domain Ω (including the complex angles up to the period), with respect to the complex current distribution U on the array. 59) with respect to the complex current distribution U on the array. 60) at a given t > 0 which describes a compromise demand of simultaneously decreasing both the approximation of a given amplitude pattern in the angular area Ω and the outgoing energy outside it (two-criterion optimization), with respect to the complex current distribution U on the array.
As a rule, such problems are reduced to homogeneous equations; the modulus of their maximal or minimal eigenvalue describes the extreme value of the objective function. g. as arguments of some complex functions). Each section contains a general formulation of the variational problem and its reduction to the Lagrange–Euler equation which is a necessary condition for the functional to be extremal. The equations are nonlinear; their solutions are, as a rule, nonunique. The simplest iterative procedures are proposed for their solution and the relaxation properties of these procedures are established.
All the functionals use the mean-square metric in the optimization criteria. This fact suggests the functional spaces in which the operators act in the problems considered. This is a class of complex functional Hilbert spaces in which, besides the usual properties of the inner product, certain additional conditions hold. We call such spaces spaces of L 2 -type. 1) for any u 1 2 H , u 2 2 H , jv j Ä C < 1. 2) is assumed to be valid for this space. 2) becomes an equality if and only if arg (u 1 ) arg (u 2 ) D c o n s t almost everywhere in support of the functions from H.
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