Sharp error estimates for a finite element-penalty approach to a class of regulator problems
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Quadratic cost optimal controls can be solved by penalizing the governing linear differential equation , . 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 angle. Under condition (H), we prove that the penalty parameter 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.