Abstract
In two and three dimensional Lipschitz, but not necessarily convex, polytopal domains, we devise and analyze a reliable and efficient a posteriori error estimator for a semilinear optimal control problem; control constraints are also considered. We consider a fully discrete scheme that discretizes the state and adjoint equations with piecewise linear functions and the control variable with piecewise constant functions. The devised error estimator can be decomposed as the sum of three contributions which are associated to the discretization of the state and adjoint equations and the control variable. We extend our results to a scheme that approximates the control variable with piecewise linear functions and also to a scheme that approximates the solution to a nondifferentiable optimal control problem. We illustrate the theory with two and three-dimensional numerical examples.
Highlights
In this work we will be interested in the design and analysis of a posteriori error estimates for finite element approximations of semilinear control–constrained optimal control problems; the state equation corresponds to a Dirichlet problem for a monotone, semilinear, and elliptic partial differential equation (PDE)
We propose an a posteriori error estimator for the optimal control problem (1.1)–(1.3) that can be decomposed as the sum of three contributions: one related to the discretization of the state equation, one associated to the discretization of the adjoint equation, and another one that accounts for the discretization of the control variable
We propose an a posteriori error estimator that accounts for the discretization of the state, adjoint state, and control variables when the error, in each one of these variables, is measured in the L2(Ω)-norm
Summary
In this work we will be interested in the design and analysis of a posteriori error estimates for finite element approximations of semilinear control–constrained optimal control problems; the state equation corresponds to a Dirichlet problem for a monotone, semilinear, and elliptic partial differential equation (PDE). To the best of our knowledge, the first work that provided an advance regarding a posteriori error estimates for linear and distributed optimal control problems is [25]: the devised residual–type a posteriori error estimator is proven to yield an upper bound for the error Theorem 3.1 of [25] These results were later improved in [21] where the authors explore a slight modification of the estimator of [25] and prove upper and lower error bounds which include oscillation terms Theorems 5.1 and 6.1 of [21]. We notice that no efficiency analysis is provided in [26] We conclude this paragraph by mentioning the approach introduced in [7] for estimating the error in terms of the cost functional for linear/semilinear optimal control problems.
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