Optimal and instantaneous control of the instationary Navier-Stokes equations

Contents Danksagung 1 Chapter 1. Introduction 5 1. Aims and scope of flow control 5 2. A brief review in active flow control 8 Chapter 2. Notation and preliminary results 15 1. Quasi-Stokes problems 15 2. Instationary Navier-Stokes equations in 2d 16 3. Navier-Stokes numerics 19 Chapter 3. Optimal control of the instationary Navier-Stokes equations 23 1. The optimal control problem 23 2. Derivatives 25 3. First order necessary optimality condition 29 4. Second order conditions 31 Chapter 4. Convergence for second order methods 35 1. Newton's algorithm 36 2. The BFGS method 38 3. Basic SQP–method 39 4. Schur–complement SQP–method 42 5. Reduced SQP–method 44 6. Reduced SQP-BFGS method 47 7. Newton's method for driven-cavity control 49 Chapter 5. Instantaneous control for finite dimensional systems 59 1. Scope of the method 59 2. The model problem 60 3. The instantaneous control strategy 61 4. Discrete and continuous output control laws 63 5. Convergence of the control laws 64 6. An example 68 3 4 CONTENTS Chapter 6. Instantaneous control for the instationary Navier-Stokes equations 71 1. Framework 71 2. Existence and uniqueness of solutions 72 3. Stability of the continuous controller 78 4. Stability of discrete controllers 80 5. Numerical validation 85 Chapter 7. Instantaneous feedback control for backward facing step flows 93 1. Problem formulation 93 2. The optimality system for (P) 95 3. Instantaneous control for boundary controls and boundary observations 97 4. Line search 100 5. Backward-facing-step numerics 103 Appendix A. A general optimal control problem, lemmata and proofs 113 1. A general control problem 113 2. Proof of Proposition 2.1 120 Bibliography 125 CHAPTER 1 Introduction 1. Aims and scope of flow control A general optimal control problem may be subdivided into the following interrelated parts [91]: 1. Problem statement Definition of the goal, cost function or performance index 2. State estimation problem Knowledge of the current state of the system 3. Modelling and system identification Knowledge of how the environment effects the past, present and future of the system 4. Optimization Determination of the best control policy based on parts 1., 2. and 3. Of course, for control problems in fluid flow this frame has a mathematical and an engineering component. From the engineering point of view it often is desirable to minimize drag, increase mixing, reduce turbulent kinetic energy and so forth in a channel, say. Engineers measure these quantities using micro-electronical devices …