On the Existence of Solutions of Two Optimization Problems
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In this paper, we prove the existence of solutions for the minimization problem of the shell weight for a given minimal frequency of the shell vibrations as well as for the maximization problem of the minimal frequency for a given shell weight. We consider an optimal control problem governed by an eigenvalue problem for a system of differential equations with variable coefficients. The form of the shell is considered as a control. Some of the coefficients are non-measurable. Earlier, we introduced certain special weighted functional spaces. By using these spaces, we establish the continuity of the considered minimal frequency functional and obtain the existence of solutions of both optimal control problems. At the end, we prove the Lipschitz continuity of the eigenvalue problem.
KeywordsEigenvalues Eigenfunctions Continuous dependence The shell of rotation Weighted spaces Non-measurable coefficients Lipschitz continuity
Mathematics Subject Classification34B09 49J15
We would like to thank the referees for their very thorough work and their many constructive comments.
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