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Sharp error estimates for a finite element-penalty approach to a class of regulator problems

Authors: Goong Chen, Wendell H. Mills, Shun Hua Sun and David A. Yost
Journal: Math. Comp. 40 (1983), 151-173
MSC: Primary 65K10; Secondary 49D30
MathSciNet review: 679438
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Abstract: Quadratic cost optimal controls can be solved by penalizing the governing linear differential equation [2], [9]. In this paper, we study the numerical analysis of this approach using finite elements. We formulate the geometric condition (H) which requires that pairs of certain related finite-dimensional approximation spaces form “angles” which are bounded away from the “$180^\circ$ angle”. Under condition (H), we prove that the penalty parameter $\varepsilon$ and the discretization parameter h are independent in the error bounds, thereby giving sharp asymptotic error estimates. This condition (H) is shown to be also a necessary condition for such independence. Examples and numerical evidence are also provided.

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Article copyright: © Copyright 1983 American Mathematical Society