Ira Neitzel

A priori error estimates for space–time finite element discretization of semilinear parabolic optimal control problems

AbstractIn this paper, a priori error estimates for space–time finite element discretizations of optimal control problems governed by semilinear parabolic PDEs and subject to pointwise control constraints are derived. We extend the approach from Meidner and Vexler (SIAM Control Optim 47(3):1150–1177, 2008; SIAM Control Optim 47(3):1301–1329, 2008) where linear-quadratic problems have been considered, discretizing the state equation by usual conforming finite elements in space and a discontinuous Galerkin method in time. Error estimates for controls discretized by piecewise constant functions in time and cellwise constant functions in space are derived in detail and we explain how error estimate for further discretization approaches, e.g., cellwise linear discretization in space, the postprocessing approach from Meyer and Rösch (SIAM J Control Optim 43:970–985, 2004), and the variationally discrete approach from Hinze (J Comput Optim Appl 30:45–63, 2005) can be obtained. In addition, we derive an estimate for a setting with finitely many time-dependent controls.

Strategies for time-dependent PDE control with inequality constraints using an integrated modeling and simulation environment

In Neitzel et al. (Strategies for time-dependent PDE control using an integrated modeling and simulation environment. Part one: problems without inequality constraints. Technical Report 408, Matheon, Berlin, 2007) we have shown how time-dependent optimal control for partial differential equations can be realized in a modern high-level modeling and simulation package. In this article we extend our approach to (state) constrained problems. “Pure” state constraints in a function space setting lead to non-regular Lagrange multipliers (if they exist), i.e. the Lagrange multipliers are in general Borel measures. This will be overcome by different regularization techniques. To implement inequality constraints, active set methods and barrier methods are widely in use. We show how these techniques can be realized in a modeling and simulation package. We implement a projection method based on active sets as well as a barrier method and a Moreau Yosida regularization, and compare these methods by a program that optimizes the discrete version of the given problem.

A Smooth Regularization of the Projection Formula for Constrained Parabolic Optimal Control Problems

We present a smooth, that is, differentiable regularization of the projection formula that occurs in constrained parabolic optimal control problems. We summarize the optimality conditions in function spaces for unconstrained and control-constrained problems subject to a class of parabolic partial differential equations. The optimality conditions are then given by coupled systems of parabolic PDEs. For constrained problems, a non-smooth projection operator occurs in the optimality conditions. For this projection operator, we present in detail a regularization method based on smoothed sign, minimum and maximum functions. For all three cases, that is, (1) the unconstrained problem, (2) the constrained problem including the projection, and (3) the regularized projection, we verify that the optimality conditions can be equivalently expressed by an elliptic boundary value problem in the space-time domain. For this problem and all three cases we discuss existence and uniqueness issues. Motivated by this elliptic problem, we use a simultaneous space-time discretization for numerical tests. Here, we show how a standard finite element software environment allows to solve the problem and, thus, to verify the applicability of this approach without much implementation effort. We present numerical results for an example problem.

Regularization for semilinear elliptic optimal control problems with pointwise state and control constraints

In this paper a class of semilinear elliptic optimal control problem with pointwise state and control constraints is studied. We show that sufficient second order optimality conditions for regularized problems with small regularization parameter can be obtained from a second order sufficient condition assumed for the unregularized problem. Moreover, error estimates with respect to the regularization parameter are derived.

A-posteriori error estimation of discrete POD models for PDE-constrained optimal control

In this work a-posteriori error estimates for linear-quadratic optimal control problems governed by parabolic equations are considered. Different error estimation techniques for finite element discretizations and model-order reduction are combined to validate suboptimal control solutions from low-order models which are constructed by a Galerkin discretization and the application of proper orthogonal decomposition. The theoretical findings are used to design an updating algorithm for the reduced-order models; the efficiency and accuracy are illustrated by numerical tests.

Finite Element Discretization of State-Constrained Elliptic Optimal Control Problems with Semilinear State Equation

We study a class of semilinear elliptic optimal control problems with pointwise state constraints. The purpose of this paper is twofold. First, we present convergence results for the finite element discretization of this problem class similarly to known results with finite-dimensional control space, thus extending results that are---for control functions---only available for linear-quadratic convex problems. We rely on a quadratic growth condition for the continuous problem that follows from second order sufficient conditions. Second, we show that the second order sufficient conditions for the continuous problem transfer to its discretized version. This is of interest, for example, when considering questions of local uniqueness of solutions or the convergence of solution algorithms such as the SQP method.

On linear-quadratic elliptic control problems of semi-infinite type

We derive a priori error estimates for linear-quadratic elliptic optimal control problems with pointwise state constraints in a compact subdomain of the spatial domain Ω for a class of problems with finite-dimensional control space. The problem formulation leads to a class of semi-infinite programming problems, whose constraints are implicitly given by the FE-discretization of the underlying PDEs. We prove an order of for the error |ū − ū h | in the controls, and show that it can be improved to an order of h 2|log h| under certain assumptions on the structure of the active set. Numerical experiments underline the proven theoretical results.

An Optimal Control Problem Governed by a Regularized Phase-Field Fracture Propagation Model

This paper is concerned with an optimal control problem governed by a regularized fracture model using a phase-field technique. To avoid the nondifferentiability due to the irreversibility constraint on the fracture growth, the phase-field fracture model is relaxed using a penalization approach. The existence of a solution to the penalized fracture model is shown, and the existence of at least one solution for the regularized optimal control problem is established. Moreover, the linearized fracture model is considered and used to establish first order necessary conditions as well as to discuss QP-approximations to the nonlinear optimization problem. A numerical example suggests that these can be used to obtain a fast convergent algorithm.

Dirichlet control of elliptic state constrained problems

We study a state constrained Dirichlet optimal control problem and derive a priori error estimates for its finite element discretization. Additional control constraints may or may not be included in the formulation. The pointwise state constraints are prescribed in the interior of a convex polygonal domain. We obtain a priori error estimates for the