Preface	6
	Acknowledgements	7
	Contents	9
	Analytical Background and Optimality Theory	12
	Introduction and Examples	12
	Introduction	12
	Examples for Optimization Problems with PDEs	15
	Optimization of a Stationary Heating Process	16
	Boundary Control	16
	Boundary Control with Radiation Boundary	17
	Distributed Control	18
	Problems with State Constraints	18
	Optimization of an Unsteady Heating Processes	18
	Boundary Control	19
	Optimal Design	19
	Linear Functional Analysis and Sobolev Spaces	20
	Banach and Hilbert Spaces	21
	Basic Definitions	21
	Linear Operators and Dual Space	22
	Sobolev Spaces	24
	Lebesgue Spaces	24
	Lebesgue Measurable Functions and Lebesgue Integral	24
	Definition of Lebesgue Spaces	26
	Density Results and Convergence Theorems	28
	Weak Derivatives	29
	Regular Domains and Integration by Parts	29
	Sobolev Spaces	30
	Poincaré's Inequality	32
	Sobolev Imbedding Theorem	33
	The Dual Space H-1 of H01	34
	Weak Convergence	35
	Weak Solutions of Elliptic and Parabolic PDEs	37
	Weak Solutions of Elliptic PDEs	37
	Weak solutions of the Poisson equation	37
	Dirichlet Boundary Conditions	37
	Boundary Conditions of Robin Type	41
	Weak Solutions of Uniformly Elliptic Equations	43
	An Existence and Uniqueness Result for Semilinear Elliptic Equations	44
	Regularity Results	45
	Interior Regularity	45
	Boundary Regularity	46
	Weak Solutions of Parabolic PDEs	47
	Uniformly Parabolic Equations	48
	Bochner Spaces	48
	Weak Solutions of Uniformly Parabolic Equations	51
	Weak Solutions	51
	Existence and Uniqueness of Weak Solutions	54
	Abstract Parabolic Evolution Problem	54
	Abstract Parabolic Evolution Problem, Equivalent Form	55
	Energy Estimate and Uniqueness Result	55
	Existence Result by Galerkin Approximation	56
	Operator Formulation	58
	Regularity Results	59
	An Existence and Uniqueness Result for Semilinear Parabolic Equations	60
	Gâteaux- and Fréchet Differentiability	61
	Basic Definitions	61
	Implicit Function Theorem	63
	Existence of Optimal Controls	63
	Existence Result for a General Linear-Quadratic Problem	63
	Existence Results for Nonlinear Problems	65
	Applications	67
	Distributed Control of Elliptic Equations	67
	Boundary Control of Semilinear Elliptic Equations	68
	Reduced Problem, Sensitivities and Adjoints	68
	Sensitivity Approach	69
	Adjoint Approach	70
	Application to a Linear-Quadratic Optimal Control Problem	71
	A Lagrangian-Based View of the Adjoint Approach	74
	Second Derivatives	75
	Optimality Conditions	76
	Optimality Conditions for Simply Constrained Problems	76
	Optimality Conditions for Control-Constrained Problems	81
	A General First Order Optimality Condition	82
	Necessary First Order Optimality Conditions	83
	Applications	84
	General Linear-Quadratic Problem	84
	Distributed Control of Elliptic Equations	85
	Distributed Control of Semilinear Elliptic Equations	87
	Boundary Control of Parabolic Equations	89
	Optimality Conditions for Problems with General Constraints	91
	A Basic First Order Optimality Condition	92
	Constraint Qualification and Robinson's Regularity Condition	93
	Karush-Kuhn-Tucker Conditions	94
	Application to PDE-Constrained Optimization	95
	Applications	97
	Elliptic Problem with State Constraints	97
	Optimal Control of Instationary Incompressible Navier-Stokes Flow	99
	Functional Analytic Setting	100
	Analysis of the Flow Control Problem	102
	Reduced Optimal Control Problem	105
	Optimization Methods in Banach Spaces	107
	Synopsis	107
	Globally Convergent Methods in Banach Spaces	109
	Unconstrained Optimization	109
	Armijo Rule	111
	Optimization on Closed Convex Sets	114
	Projected Armijo Rule	117
	General Optimization Problems	119
	Newton-Based Methods-A Preview	119
	Unconstrained Problems-Newton's Method	119
	Simple Constraints	120
	Nonsmooth Reformulation Approach and Generalized Newton Methods	120
	SQP Methods	122
	General Inequality Constraints	123
	Nonsmooth Reformulation Approach and Generalized Newton Methods	124
	SQP Methods	125
	Generalized Newton Methods	125
	Motivation: Application to Optimal Control	125
	A General Superlinear Convergence Result	126
	The Classical Newton's Method	129
	Generalized Differential and Semismoothness	130
	Semismooth Newton Methods	133
	Semismooth Newton Method for Finite Dimensional KKT Systems	134
	Discussion	135
	Semismooth Newton Methods in Function Spaces	135
	Pointwise Bound Constraints in L2	135
	Semismoothness of Superposition Operators	136
	Pointwise Bound Constraints in L2 Revisited	139
	Application to Optimal Control	140
	General Optimization Problems with Inequality Constraints in L2	142
	Application to Elliptic Optimal Control Problems	143
	Distributed Control	143
	Neumann Boundary Control	145
	Optimal Control of the Incompressible Navier-Stokes Equations	147
	Sequential Quadratic Programming	150
	Lagrange-Newton Methods for Equality Constrained Problems	150
	The Josephy-Newton Method	154
	Generalized Equation	154
	SQP Methods for Inequality Constrained Problems	158
	SQP Subproblem	160
	Application to Optimal Control	161
	State-Constrained Problems	161
	SQP Methods	162
	Semismooth Newton Methods	162
	Moreau-Yosida Regularization	163
	Lavrentiev Regularization	164
	Further Aspects	165
	Mesh Independence	165
	Application of Fast Solvers	166
	Other Methods	166
	Discrete Concepts in PDE Constrained Optimization	167
	Introduction	167
	Control Constraints	168
	Stationary Model Problem	168
	First Discretize, Then Optimize	170
	First Optimize, Then Discretize	171
	Discussion and Implications	173
	The Variational Discretization Concept	174
	Error Estimates	177
	Uniform Estimates	180
	Numerical Examples for Distributed Control	181
	Boundary Control	187
	Neumann and Robin-Type Boundary Control	188
	Numerical Examples for Robin-Type Boundary Control	193
	Dirichlet Boundary Control	199
	Numerical Example for Dirichlet Boundary Control	203
	Some Literature Related to Control Constraints	206
	Constraints on the State	207
	Pointwise Bounds on the State	208
	Finite Element Discretization	209
	Error Analysis	213
	Piecewise Constant Controls	217
	Numerical Examples for Pointwise Constraints on the State	222
	Some Literature for (Control and) State Constraints	228
	Pointwise Bounds on the Gradient of the State	229
	Finite Element Discretization	231
	A Numerical Experiment with Pointwise Constraints on the Gradient	236
	Time Dependent Problem	237
	Mathematical Model, State Equation	237
	Optimization Problem	239
	Discretization	239
	Further Literature on Control of Time-Dependent Problems	241
	Applications	243
	Optimal Semiconductor Design	243
	Semiconductor Device Physics	244
	Charge Transport	245
	The Potential Equation	246
	The Continuity Equations	247
	The Current Densities	248
	Scaling	249
	The Optimization Problem	250
	The First-Order Optimality System	254
	Numerical Results	256
	Steepest Descent	256
	The Reduced Newton Method	258
	Optimal Control of Glass Cooling	260
	Modeling	261
	Radiation	261
	SPN-approximations	263
	Optimal Boundary Control	264
	Derivatives	267
	Newton's Method	268
	Numerical Results	270
	References	274