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Decoupling and Optimization of Differential-Algebraic Equations

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Decoupling and Optimization of Differential-Algebraic Equations Decoupling and Optimization of Differential-Algebraic Equations with Application in Flow Control vorgelegt von Dipl.-Math.Techn. Jan Heiland geb. in Friedrichshafen Von der Fakultät II - Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften – Dr. rer. nat. – genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. Rolf Möhring Gutachter: Prof. Dr. Michael Hinze Gutachter: Prof. Dr. Volker Mehrmann Gutachter: Prof. Dr. Tomáš Roubíček Tag der wissenschaftlichen Aussprache: 13. Februar 2014 Berlin 2014 D 83 1 Contents 1. Introduction 3 2. Preliminary Notions and Notations 7 2.1. A class of Semi-explicit Semi-linear DAEs 7 2.2. The Index of the DAEs 7 2.3. The Index of DAEs with Inputs 10 2.4. Functional Analysis Framework 11 3. Decoupling of Semi-linear Semi-explicit Index-2 ADAEs 19 3.1. Semi-explicit Semi-linear ADAEs of Index 2 19 3.2. Decoupling of the Equations 21 3.3. Decomposition of the Solution 26 3.4. Decoupling of the System 26 3.5. Application to the Navier-Stokes Equation 34 3.6. Application to the Maxwell Equation 37 4. Semi Discretization of the State Equations 41 4.1. Galerkin Approximation 41 4.2. Decomposition of the Discrete Solutions 44 4.3. Convergence of the (External) Approximation Scheme 47 4.4. Convergence of the Galerkin Approximations 49 4.5. Initial Conditions 59 5. Constrained Optimization and Optimal Control 61 5.1. Multipliers and First Order Necessary Optimality Conditions 61 5.2. Relation to Optimal Control of Systems and the Adjoint State 62 5.3. First Order Sufficient Optimality Conditions 63 6. Optimal Control Problem and the Adjoint Equation 67 6.1. Semi-discretization of the Adjoint Equation 71 6.2. Optimal Control of the Semi-discrete Equations 77 7. Iterative Solution of the Nonlinear Optimality System 81 7.1. Linearizations and Newton Schemes for the Functional Equations 82 7.2. Newton for the Optimality Conditions 83 7.3. Newton for the Reduced Cost Functional 87 8. Optimal Control of Finite-dimensional Index-2 DAEs 89 8.1. The Finite-dimensional State Equations 90 8.2. Optimal Control of Semi-explicit DAEs 92 8.3. Linear-quadratic Optimal Control 93 8.4. Existence and Representations of Optimal Solutions 95 8.5. Crossterms and the Algebraic Variable in the Cost Functional 103 8.6. Optimal Control Including the Algebraic Variables 104 9. Numerical Algorithms and Application Example 109 9.1. Linear-Quadratic Setup and Algorithms 109 9.2. Solution of Constrained Lyapunov Equations 113 9.3. Distributed Control of a Driven Cavity 117 10. Discussion and Outlook 121 Acknowledgment 123 Index 125 References 127 2 3 1. Introduction The notion of the state is used to describe all kinds of technical, physical, chem- ical, sociological, or mental phenomena. The state can mean a velocity, a temper- ature, a color, a noise, or just a state of mind. The state can vary with the time, with the location, or with other parameters. For states that vary with time and location, one can use partial differential equations to model the underlying phenomena in mathematical terms [21, 22]. Then one can analyze the obtained equations, in order to understand or also to control the phenomena. In view of employing computers for understanding or controlling, one approxi- mates the continuous equations by discrete equations. This is called a discretization. In practice, it is generally recognized that finite-dimensional or discrete formu- lations are well suited to deliver approximate solutions to continuous or infinite- dimensional problems. Thus one can use common numerical methods, optimized to solve the finite-dimensional equations, and to some extent forget about the actual aim of learning about the infinite-dimensional phenomena. The design of discretizations that preserve certain properties of the continuous equations, like positivity of the solution, dissipassivity, symmetry, or energy of the system, has been proven very successful in providing efficient and reliable approxi- mations. It may happen that the discretization introduces properties that the continuous system does not have. Consider the example of the Navier-Stokes Equation that model the state of an incompressible flow via its velocity v and its pressure p in a domain Ω and a time interval (0,T ). The describing equations are given via the system v˙ + (v · ∇)v + ∇p − ν∆v = f, (1.1a) div v = 0, in Ω × (0,T ), (1.1b) and v|t=0 = α and v|∂Ω = γ, (1.1c) consisting of the momentum equation with a viscosity parameter ν, the constraint that the flow is divergence-free, and a condition on the initial state of the velocity plus values for the velocity at the boundary ∂Ω. Discretizing the spatial component in (1.1), i.e. approximating v(t) and p(t) via finite-dimensional vectors vk(t) and pk(t), a discrete approximation to (1.1) is typically given as T Mv˙k − A(vk) − Jk pk = fk, (1.2a) Jkvk = 0, in (0,T ) (1.2b) and vk(0) = αk. (1.2c) Here, k is a parameter describing the order of approximation, and the matrices Jk T and Jk represent the differential operators div and ∇ in finite dimensions. For all common discretizations, the so called mass matrix M is invertible, so that from Equations (1.2a-c) one can infer, that if Jkαk = 0, then a solution vk fulfills Jkv˙k = 0 for all time. Numerical methods [50, 144], that are state of the art, use the property that (1.2) implicitly defines the Pressure Poisson Equation −1 T −1 −1 −JkM Jk pk = JkM f + JkM A(vk)k, (1.3) to decouple the computations of vk and pk. 4 However, the basic assumption that Jkv˙k = 0 will not necessarily transfer to the continuous case of (1.1), since in the general formulation v˙ has to be assumed of low regularity and divv ˙ is not defined. The proof whether or not there a solution possesses a higher regularity is a key in the quest for a solution to the millennium problem addressing the existence and uniqueness of solutions to the Navier-Stokes Equation [103]. In our considerations, the use of (1.3) is legitimate in finite di- mensions but it might not be a proper approximation in the limit case, where the discretization is arbitrarily close to the continuous equation, cf. also the remarks in [50, p. 642]. Other examples relate to system theoretic properties like controllability and the question when their presence in the discrete equations is transferred to the limit case, see [105, 108] and see [77] for an illustrating example where this is not the case. The above example of the Pressure Poisson Equation points to the general issue about the conditions under which a transformation of the discrete approximation commutes with a discrete approximation of a related transformation of the continu- ous problem. In this thesis on decoupling and optimization of differential-algebraic equations, we consider two such transformations, namely (a) formulation of the optimal control problem as a system of optimality con- ditions and (b) decoupling of the differential and algebraic equations and their interaction with numerical approximations. We will investigate when do these transformations, whose finite-dimensional counterparts are well understood, also apply in the continuous setting and whether they commute with discretizations. As for optimal control, the question of whether to optimize or to discretize first has been investigated in all fields of application, cf. [79] and see [33, 55, 146, 147] for examples in optimal flow control. As for infinite-dimensional differential-algebraic equations, the interaction of decoupling and discretization has attained attention recently [6, 7, 41]. Understanding the transformations in infinite dimensions may lead to discretiza- tions that come with properties that are desired and compliant with those of the continuous equation. As an example consider the skew-symmetrization of the con- vective term in weak formulations of the Navier-Stokes Equation such that also the discretized convection is skew-symmetric, [68, Ch. 3]. Also, the direct application of algorithms for model reduction [63, 138, 123], update formulas in optimization [52, 72, 78, 95], or decoupling of differential and algebraic parts [6, 7] to infinite dimensional systems have been proven successful for numerical approximation. The idea of interchanging transformations like discretization, optimization and decoupling is also reflected in the second part of the thesis dealing with the opti- mal control of finite-dimensional of differential-algebraic equations. Instead of first formulating a decoupling and then stating optimality conditions, we set up formal optimality conditions for the original system. Thus, a possibly necessary decoupling can be applied with respect to efficient numerical approximation rather than being used for theoretical considerations. We use the particular structure of the equations to show that the solution of the original system implicitly leads to the solution of an equivalent system for which differential and algebraic parts are partially decoupled and optimality conditions are well understood. Staying with the original differential-algebraic equations and the native variables is motivated by practical considerations. In computations, the decoupling, for ex- ample the restriction to the space of divergence-free functions in the Navier-Stokes setting, is not advisable in general or simply unfeasible, cf. the discussion in [7]. 5 Also, if the optimality conditions are in the form of the state equations, then one can resort to the same solvers. The thesis is structured as follows. We start with an introduction of notions related to differential-algebraic equations and of the functional analysis framework needed to treat infinite-dimensional equations. In Section 3, we define a class of infinite-dimensional differential-algebraic equations and derive a decoupling of dif- ferential and algebraic parts.
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