Numerical analysis for the approximation of optimal control problems with pointwise observations

Abstract

Communicated by Q. WangIn this paper, we study the numerical methods for optimal control problems governed by elliptic PDEs with pointwise observations of the state. The first order optimality conditions as well as regularities of the solutions are derived. The optimal control and adjoint state have low regularities due to the pointwise observations. For the finite dimensional approximation, we use the standard conforming piecewise linear finite elements to approximate the state and adjoint state variables, whereas variational discretization is applied to the discretization of the control. A priori and a posteriori error estimates for the optimal control, the state and adjoint state are obtained. Numerical experiments are also provided to confirm our theoretical results.of elliptic equations with measure data presented in [16] and [17], a priori error estimates for the control, the state, and the adjoint state are derived. Moreover, the a posteriori error estimates of residual type are presented, which can be applied as indicators for the mesh refinement in adaptive finite element methods. Numerical examples are also included to illustrate the theoretical results. It should be pointed out that our theoretical and numerical results are still valid when the penalty parameter˛is very small, which often appears in many applications, for example, inverse problems.Our paper is organized as follows. In the next section, we present the mathematical setting and formulate the optimal control problems in detail. Finite element approximation to optimal control problems is also presented. In Sections 3 and 4, we prove the a priori error and a posteriori error estimates, respectively. We finally present numerical results, which confirm our analytical findings in section 5.where B is the linear operator from U to L 2 . /, U is the control space. Here, B can be an identity operator such that Bu D u or Bu D ! u where ! is an open subset of and ! the characteristic function of the subset !. z D .z 1 , , z m / 2 R m is given, and X j (j D 1, , m) are space points strictly contained in . Moreover, U ad U is a closed convex subset of typeAccording to the assumptions on and A, for each Bu 2 L 2 . /, we have y.x; u/ 2 H 2 . / \ H 1 0 . /, thus y.x; u/ 2 C. / by imbedding theorem and the aforementioned optimal control problem is well-defined. From standard arguments ([18]), we can prove that the

A posteriori error estimates for the Stokes problem with singular sources

“…The motivation of the previous paragraph suggests that we seek for solutions of (1) in the weighted spaces defined in Section 2. To accomplish this task, we define the bilinear forms (11) a :…”

Pointwise error estimates for a generalized Oseen problem and an application to an optimal control problem

“…For Ω ⊂ R 2 being such that (6) and (7) holds, letting (G i , i ) be the solution of any of the regularized problems (26), (30), or (31)…”

An Adaptive FEM for the Pointwise Tracking Optimal Control Problem of the Stokes Equations

“…The pointwise tracking optimal control problem for the Poisson equation has been considered in a number of works [8,9,12,14]. In [8], the authors operate under the framework of Muckenhoupt weighted Sobolev spaces [43] and circumvent the difficulties associated with the underlying adjoint equation: a Poisson equation with a linear combination of Dirac deltas as a forcing term.…”

“…In [8], the authors operate under the framework of Muckenhoupt weighted Sobolev spaces [43] and circumvent the difficulties associated with the underlying adjoint equation: a Poisson equation with a linear combination of Dirac deltas as a forcing term. Weighted Sobolev spaces allow for working under a Hilbert space-based framework in comparison to the non-Hilbertian setting of [9,12,14]. An priori error analysis for a standard finite element approximation of the aforementioned problem can be found in [8,12] while its a posteriori error analysis has been recently provided in [4,14].…”

“…Weighted Sobolev spaces allow for working under a Hilbert space-based framework in comparison to the non-Hilbertian setting of [9,12,14]. An priori error analysis for a standard finite element approximation of the aforementioned problem can be found in [8,12] while its a posteriori error analysis has been recently provided in [4,14]. In contrast to these advances and to the best of our knowledge, the pointwise tracking optimal control problem for the Stokes equations has not been considered before.…”