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The Approximation of the Optimal Control of Vibrations—A Geometrical Approach
Author(s) -
Read John
Publication year - 1996
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(19960125)19:2<87::aid-mma761>3.0.co;2-6
Subject(s) - mathematics , banach space , duality (order theory) , optimal control , convergence (economics) , operator (biology) , norm (philosophy) , hilbert space , mathematical analysis , mathematical optimization , pure mathematics , biochemistry , chemistry , repressor , political science , transcription factor , law , economics , gene , economic growth
Abstract In this paper we employ concepts from Banach space geometry in order to examine the problem of approximating the optimal distributed control of vibrating media whose motion is governed by a wave equation with a 2 n ‐order self‐adjoint and positive‐definite linear differential operator. We show that this geometrical approach, arrived at via duality theory, provides the exact framework in which the approximation problem must be placed in order to get the correct convergence results, for it is here that the necessary and sufficient conditions for the approximate norm or time minimal control can be fully developed. Using the theory of Asplund, we are also able to improve the traditional weak * convergence results for the more difficult case of L ∞ controls. Finally, we consider certain numerical examples which help illustrate our theoretical results.