Solving Time-Dependent Optimal Control Problems in Comsol Multiphysics by Space-Time Discretizations

We use COMSOL Multiphysics to solve time-dependent optimal control problems for partial differential equations whose optimality conditions can be formulated as a PDE. For a class of linear-quadratic model problems we summarize known analytic results on existence of solutions and first order optimality conditions that exhibit the typical feature of time-dependent control problems, namely the fact that a part of the optimality system has to be integrated backward in time. We present a strategy that is based on the treatment of the coupled optimality system in the space-time cylinder. A brief motivation of this approach is given by showing that the optimality system is elliptic in some sence. Numerical examples show advantages and limits of the usage of COMSOL Multiphysics and of our approach

The convergence of an interior point method for an elliptic control problem with mixed control-state constraints.

The paper addresses primal interior point method for state constrained PDE optimal control problems. By a Lavrentiev regularization, the state constraint is transformed to a mixed control-state constraint with bounded Lagrange multiplier. Existence and convergence of the central path are established, and linear convergence of a short-step pathfollowing method is shown. The behaviour of the regularizations are demonstrated by numerical examples

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

We present a smooth, i.e. 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, i.e (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 implementational effort. We present numerical results for an example problem

Application of Proper Orthogonal Decomposition Methods in Reactive Pore Diffusion Simulations

Reactive pore diffusion is an important process in automotive exhaust-gas aftertreatment modelling the overall conversion of pollutants. It features highly nonlinear source terms from chemical reactions coupled with transport processes. This work examines the application of model reduction by proper orthogonal decomposition. It is shown that this technique can be successfully applied to the system using separate bases for each species. Using a basis obtained for baseline conditions, predictions can be made for species profiles within a pore system for different conditions, potentially leading to significantly reduced computational requirements

Solving optimal PDE control problems : optimality conditions, algorithms and model reduction

This thesis deals with the optimal control of PDEs. After a brief introduction in the theory of elliptic and parabolic PDEs, we introduce a software that solves systems of PDEs by the finite elements method. In the second chapter we derive optimality conditions in terms of function spaces, i.e. a systems of PDEs coupled by some pointwise relations. Now we present algorithms to solve the optimality systems numerically and present some numerical test cases. A further chapter deals with the so called lack of adjointness, an issue of gradient methods applied on parabolic optimal control problems. However, since optimal control problems lead to large numerical schemes, model reduction becomes popular. We analyze the proper orthogonal decomposition method and apply it to our model problems. Finally, we apply all considered techniques to a real world problem.:Introduction The state equation Optimal control and optimality conditions Algorithms The \"lack of adjointness\" Numerical examples Efficient solution of PDEs and KKT- systems A real world application Functional analytical basics Codes of the example

The Interior Point Method for the solution of optimal control problems subject to PDE and point-wise state constraints

In der vorliegenden Arbeit wird die Innere-Punkte-Methode für die optimale Steuerung partieller Differentialgleichung mit gegebenen Beschränkungen an den Zustand untersucht. Als zu minimierendes Funktional betrachten wir ein “tracking-type” Funktional, die Gleichungsnebenbedingung ist eine instationäre lineare partielle Differentialgleichung und die Beschränkungen an den Zustand sind einfache Box-Beschränkungen. Die Zustandsbeschränkungen werden eliminiert, indem sie zuerst mit der Lavrentiev-Technik regularisiert und dann mittels parametrisierter logarithmischer Strafterme in das Funktional integriert werden. In der vorliegenden Arbeit wird gezeigt, dass für jeden festen Parameterwert das Innere-Punkte-Problem eine eindeutige Lösung besitzt. Diese ist strikt zulässig. Es wird die Konvergenz der Lösungen gegen die Lösungen der regualrisierten Aufgabe bewiesen. Ausgehend davon wird ein Algorithmus im Funktionenraum konstruiert, der ein Newton-Verfahren mit gleichzeitiger Verringerung des Pfadparameters kombiniert. Die Konvergenz dieses Algorithmus wird bewiesen. Ausgehend von dem im Funktionenraum konstruierten Algorithmus wird eine Implementierung der Innere-Punkte-Methode vorgenommen. Dabei wird besonderer Wert auf die Integration schon vorhandener Software zur Lösung partieller Differentialgleichungen gelegt. Daraus resultiert die Implementierung einer Matlab-Klasse, die wesentliche Teile der PDE-Toolbox benutzt. Ein weiterer Aspekt bei der Implementierung der Innere-Punkte-Methode ist, inwieweit sich die Methode in bestehende Software integrieren lässt. Dabei zeigt sich, dass sich mittels der Innere-Punkte-Methode eine sehr leicht zu implementierende Optimalsteuersoftware auf Basis des Programmpakets COMSOL Multiphysics realisieren lässt. Dabei muss lediglich die Verringerung des Pfadparameters implementiert werden, den Newtonschritt mitsamt adaptiver Gitterverfeinerung übernimmt der COMSOL-Löser. Eine Reihe numerischer Beispiele bestätigt die theoretischen Konvergenzaussagen.In this work we investigate the interior-point method for the optimal control subject to partial differential equations and point-wise state constraints. The objective functional is of “tracking-type”, the PDE is linear parabolic and the state constraints are from the box-type. We regularize the state constraints by Lavrentiev-regularized and eliminate them by a parametrized logarithmic barrier terms. We show that for every fixed parameter the interior point problem has a unique solution. It is strictly feasible. We show the convergence of the solutions of the parametrized problem towards the solutions of the regularized problem. Based on this we develop an algorithm in function spaces that combines Newton's method with the reduction of the path parameter. We prove the convergence of this algorithm. The next step is the implementation of the IP-method by using standard tool which leads to a OO code that uses some of Matlab's PDE toolbox functionality. Further, we integrate the IP-method in existing PDE software like COMSOL Multiphysics. Some numerical examples confirm the (theoretical proved) rate of convergence

Coupled

This paper deals with the numerical solution of optimal control problems with control complementarity constraints. For that purpose, we suggest the use of several penalty methods which differ with respect to the handling of the complementarity constraint which is either penalized as a whole with the aid of NCP-functions or decoupled in such a way that non-negativity constraints as well as the equilibrium condition are penalized individually. We first present general global and local convergence results which cover several different penalty schemes before two decoupled methods which are based on a classical ℓ1- and ℓ2-penalty term, respectively, are investigated in more detail. Afterwards, the numerical implementation of these penalty methods is discussed. Based on some examples, where the optimal boundary control of a parabolic partial differential equation is considered, some quantitative properties of the resulting algorithms are compared