Robust Optimal Filtering and Argument Transforming for Feedback Controlling Optical Systems

摘要

A profound mathematical study has been accomplished for the problem of optimal discrete Fourier filtering: existence and uniqueness theorems both for direct and conjugate initial-boundary value problems for a functional differential parabolic equation have been proved; the solvability of the optimal filtering problem has been demonstrated; a functional gradient formula has been obtained. A projection finite-element scheme has been developed both for the direct and conjugate problems. A conditional gradient iterative procedure has been developed and applied for finding the optimal Fourier filter to control the feedback optical system. The optimization performance has been analyzed depending on the parameters of the optical system and on the number of the control Fourier channels. A mathematical statement of the optimal control problem has been developed for a distributed spatial argument transform. It is applied to studying controlling optical systems with nonlocal transforms of the light wave in the feedback loop. The approach developed utilizes a generalized way of the determination of the argument transform. As an advantage, one can use wide range of nonsmooth and irreversible argument transforms, and consequently achieve better results with the optimal control. The approach developed forms the basis for constructing a projection finite-element method of approximation of both direct and conjugate problems in the same manner. As a result, computational versions of projection gradient and conditional gradient methods have been worked out for target functional minimization.