Sharp Error Estimates for a Finite Element-Penalty Approach to a Class of Regulator Problems

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.