Optimal Control of Partial Differential Equations I+II

Transcription

1 About the lecture: Optimal Control of Partial Differential Equations I+II Prof. Dr. H. J. Pesch Winter semester 2011/12, summer semester 2012 (seminar), winter semester 2012/13 University of Bayreuth Preliminary Remark. This two semester course and the seminar is suited for all students of mathematical curricula from the fifth semester (BA), resp. the first semester (MA). This course can also be part of the education within the mathematical doctoral programmes of the Bayreuther Graduiertenschule or the international graduate program Identification, Optimization and Control with Applications in Modern Technologies in the framework of the Bavarian Elite Network; see www2.am.uni-erlangen.de/elitenetzwerkoptimierung/index en.php. The seminar inbetween the two courses part I and II mainly deals with numerical methods, while the first part is devoted to the theory of optimal control problems with elliptic partial differential equations (Chapters 0, 1, 2, and 4: ), the second with parabolic partial differential equations (Chapters 3, 5: , 5.7, 5.8., and 7. These courses are rounded off by guest lectures of distinguished researchers on this field and by seminars on research reports of the members of the Chair of Mathematics in Engineering Sciences. Interested students can go about doing master theses after course I. Summary. Owing to its importance for engineering applications, the field of PDE constrained optimization including optimal control of partial differential equations (PDEs) has become increasingly popular. In the near future, mathematical optimization methods will be able to solve problems whose complexity has so far allowed only the application of simulation-based methods. Hence, there is a strong need for new efficient methods for PDE constrained optimization which are capable of tackling real-life engineering applications constituting some of today s major challenges in applied mathematics. Without doubt Lion s book [3] is still the standard for optimal control problems with linear equations and convex functionals. Nonconvex problems with semilinear equations of elliptic and parabolic type are in the focus of Tröltzsch s recent book [4] particularly concerning questions of existence of solutions and optimal controls, the derivation of necessary conditions and adjoint equations as well as of second order sufficient conditions. Especially Tröltzsch s book will provide the basis for this course. Hence, we will concentrate on the theory of optimal control for elliptic (part 1) and parabolic (part 2) equations. Optimal control theory of hyperbolic equations will not be discussed. It would be beyond the scope of these courses. Although numerical methods will not be in the main focus of the lecture course, some main algorithmic ideas of numerical concepts will be described. Therefore, a seminar will be inserted which will cover modern concepts of numerical methods along the lines of the book of Hinze et. al. [2]. See also the new book of Borzi and Schulz [1]. Main

2 Literature: Borzi, A., Schulz, V.: Computational Optimization of Systems Governed by Partial Differential Equations. Philadelphia: SIAM, Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Berlin: Springer, Lions, J. L.: Optimal Control of Systems Governed by Partial Differential Equations. Berlin: Springer, Tröltzsch, F.: Optimalsteuerung bei partiellen Differentialgleichungen. Wiesbaden: Vieweg, 2. Auflage, English Translation: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Graduate Studies in Mathematics, AMS Additional references will be given on the assignement sheets. Organisation of lecture courses: Version 1: In the 1st cycle from summer semester 2006 summer semester 2007, in the 2nd cycle from summer semester 2008 winter semester 2008/09, and in the 3rd cycle from winter semester 2009/10 summer semester 2010, the lecture course was organized according to the following outline. Version 2: In the 4th cycle in winter semester 2011/12 and winter semester 2012/13 with a seminar in-between, the lecture course was organized as follows: Part 1: Chapters 0, 1, 2, and , and 4.9. Part 2: Chapters 3, 5.1 5,7, 5.8, and 7. The numbering of the exercises belong to Version 2 with footnote according Version 1.

3 0. Historical Remarks 4 The pioneers of optimal control theory Optimal control of ordinary differential equations Optimal control of partial differential equations 1. Introduction Examples for convex problems Optimal stationary heating Optimal boundary temperature First discussion Optimal temperature source Optimal instationary heating Optimal vibrating 1.2 Examples for non-convex problems Problems with semilinear elliptic equations Heating by thermal radiation Ginzburg-Landau equation in supraconductivity Control of stationary currents Problems with semilinear parabolic equations s Instationary fluid flow control 1.3 Classification of 2nd order partial differential equations Formal classification Definitions: Discriminant of a partial differential equation, elliptic, parabolic, hyperbolic Standard examples and others Non-formal classification of 2nd order partial differential equations Necessity of a classification Definitions: elliptic, parabolic, hyperbolic s and characteristic curves Suitable initial and boundary conditions Assignment No 1: Classification of PDEs 12

4 1.4 Basic concepts of optimization in the finite dimensional case Finite dimensional problems of optimal control Reformulation Existence of optimal controls Definition: optimal control, optimal state Theorem 1.1: Existence theorem First order necessary conditions Idea Notations Theorem 1.2: Variational inequality Adjoint state Idea Definition: adjoint equation Simplification of gradient and directional derivative by means of the adjoint state Theorem 1.3: Unique solution of the adjoint equation, variational inequality First main result: Optimality system (necessary conditions) Lagrange function Aim Definition: Lagrange function Reformulation of the optimality system Discussion of the variational inequality Discussion Formulation as Karush-Kuhn-Tucker system Eliminating inequality constraints Theorem 1.4: Existence of Lagrange multipliers, optimality system Definitions: (strongly) active inequality constraints 2 Critical cone 3 Theorem 1.5: Sufficient condition 3 Outlook on pde constrained optimization 2 presented in Chapter not presented in course

5 2. Linear quadratic elliptic control problems Normed vector spaces 3 Normed linear spaces Convergence of sequences, Cauchy sequences Banach spaces Pre-Hilbert spaces Hilbert spaces 2.2 Sobolev spaces L p spaces Definition: L p spaces Definition: L spaces Notations: domain, closure, boundary of a domain Definition: support Notations: multi-index, partial derivatives Definition: C k spaces Regular domains Definition: domains of class C k,1 Introduction of a Lebesgue measure on the boundary of a domain Weak derivatives and Sobolev spaces Formula of partial integration Definition: locally integrable Definition: weak derivative Definition: Sobolev spaces, Sobolev norms Definition: Closure, dense, Sobolev spaces with zero boundary conditions Theorem 2.1: Trace theorem Definition: Trace operator, trace Assignment No 2: Sobolev spaces Weak solution of elliptic equations Poisson equation Variational formulation Definition: Weak solution, variational formulation Generalizations Lemma 2.2: Theorem of Lax-Milgram Lemma 2.3: Friedrichs inequality Theorem 2.4: Existence and uniqueness of the solution of the Poisson equation

6 2.3.2 Boundary conditions of third kind See exercises Differential operators in divergence form See exercises Assignment No 3: Theorems of Lax-Milgram and Riesz 9 Assignment No 4: Trace Lemma 12 Assignment No 5: Elliptic boundary value problems in divergence form (add on) Linear mappings Linear continuous operators and functionals Definitions: linear, continuous, bounded operators, resp.functionals Theorem 2.5: Equivalence between boundness and continuity of linear operators Definitions: norm of an operator, space of all linear and continuous operators, dual space, norm of the dual space Theorem 2.6: Riesz representation theorem Definitions: bidual space, canonical embedding, reflexive spaces Assignment No 6: Linear Operators, linear Functionals, L p spaces Weak convergence Definitions: weakly convergent, weak limit point Some results on weakly convergent sequences of a weakly convergent sequence with limit point zero, but all elements lying on the unit sphere of the respective Hilbert space Some results for the application of the concept of weak convergence Definitions: weakly sequentially continuous, weakly sequentially closed, relatively weakly sequentially compact, weakly sequentially compact s for weakly sequentially continuous and non continuous operators (functionals) Consequences and conclusions for latter applications Theorem 2.7: Every bounded set of a reflexive Banach space is relatively weakly sequentially compact Theorem 2.8: Every convex and closed set is weakly sequentially closed Definitions: convex functionals, weakly lower semi-continuous functionals Theorem 2.9: Every convex and continuous functional on a Banach space is weakly lower semicontinuous : norm Assignment No 7: Weak convergence and related terms 8

7 2.5 Existence of optimal controls Optimal stationary temperature control problem with a distributed control : optimal stationary temperature control problem with a distributed control General assumptions of the chapter Choice of control space Set of admissible controls Application of Theorem 2.4 Definitions: optimal control, optimal state Definition: the control-state operator A quadratic optimization problem in Hilbert space Theorem 2.10: Existence and uniqueness of an optimal solution of the quadratic optimization problem (1st version) Theorem 2.11: Existence and uniqueness of an optimal solution of the model problem (1st version) One-sided open admissible sets Theorem 2.12: Existence and uniqueness of an optimal solution of the quadratic optimization problem (2nd version) Theorem 2.13: Existence and uniqueness of an optimal solution of the model problem (2nd version) : optimal stationary temperature source with prescribed ambient temperature Results on existence and uniqueness Optimal stationary temperature control problem with a boundary control A modified model problem: optimal stationary temperature control problem with a boundary control Theorem 2.14: Existence and uniqueness of an optimal solution General elliptic equations and functionals A general optimal control problem General assumptions Existence and uniqueness of an optimal solution 2.6 Differentiablity in Banach spaces Gâteaux derivative Definitions: directional derivative, first variation, Gâteaux derivative : The Gâteaux derivative need not be linear s: among others: squared norm Fréchet derivative Definitions: Fréchet differentiable, Fréchet derivative s: among others: Fréchet derivative of linear operators Theorem 2.15: Fréchet differentiability implies Gâteaux differentiablity Remark and Example: Counterexample that the reverse does not hold Chain rule

8 Assignment No 8: Gâteaux and Fréchet differentiability 6 Assignment No 9: Calculus of variations (add on) Adjoint operators 4 Motivation for the adjoint operator Definition: dual operator in Banach spaces Continuity of the dual operator Notation: duality pairing Definition: adjoint operator in Hilbert spaces : an integral operator 2.8 Necessary first order optimality conditions Quadratic optimization problems in Hilbert spaces Problem statement (repetition) Lemma 2.16: A variational inequality as necessary condition Remarks Lemma 2.17: A variational inequality as sufficient condition Theorem 2.18: Necessary and sufficient condition for the quadratic optimization problem in Hilbert space Rewriting of the variational inequality without the adjoint operator Optimal stationary temperature control problem with a distributed control Problem statement (repetition) Determining the adjoint operator Lemma 2.19: Auxiliary lemma Lemma 2.20: Determination of the adjoint operator Remarks Adjoint state and optimality system Definition: adjoint equation, adjoint state Theorem 2.21: Necessary and sufficient condition Optimality system Pointwise discusion of the optimality conditions Lemma 2.22: Pointwise variational inequality Theorem 2.23: Minimum principles, sufficient conditions The weak minimum principle The minimum principle Corollary 2.24 to the weak minimum principle Consequences of the Corollary Theorem 2.25: Projection formula for the optimal control The unconstrained case Formulation as Karush Kuhn Tucker system Theorem 2.26: On complementarity conditions The Karush Kuhn Tucker form of the optimality system Definition: Lagrange multipliers

9 The gradient of the objective function Lemma 2.27: On the gradient of the objective function Rewriting of the variational inequality using the gradient Stationary temperature source and boundary conditions of third kind Problem statement Adjoint equation, optimality condition Optimal stationary temperature control problem with a boundary control Problem statement Adjoint equation, optimality condition Theorem 2.28: Necessary conditions Discussion of the variational inequality Theorem 2.29: Minimum principle and projection formula for the optimal control The two cases: λ = 0 and λ > A linear optimal control problem Problem statement Optimality conditions Assignment No 10: Optimality conditions Construction of test problems Bang-bang control Problem statement with free parameter functions Construction of an analytical solution Distributed control and Neumann boundary conditions Problem statement with free parameter functions Construction of an analytical solution 2.10 The formal Lagrange principle 7 Idea Exact versus formal Lagrange principle Procedures up to now versus from now on : Optimal stationary boundary control Discussion of mathematical inaccuracies Lagrange s optimization problem Definition: Lagrange function Discussion of the formal Lagrange principle Assignment No 11: The formal Lagrange principle Numerical methods The Conditional gradient method Formulation of the conditional gradient method in Hilbert spaces Algorithm Application to elliptic problems Discussion

10 Transcription method Transformation of the infinite dimensional problem into a finite dimensional problem Disussion Transcription method: reduced form Discussion of the computational effort for the non-reduced form Establishing the reduced form Discussion of the reduced form Treatment of the reduced form Discussion Active set strategies The infinite dimensional case Primal-dual active-set strategy Discussion Algorithm Discussion The finite dimensional case Primal-dual active-set algorithm 2.12 Some final remarks and ideas 3 The adjoint state as multiplier Higher regularity of solutions Assignment No 12: Optimal control of ordinary differential equations (add on) 1 16 Assignment No 13: Novel problems I 2 16 Assignment No 14: Novel problems II new in WS 2011/12 2 new in WS 2011/12 3 new in WS 2011/12

11 3. Linear quadratic parabolic control problems Introduction 6 Elliptic versus parabolic problems Road map Derivation of optimality conditions by the formal Lagrange ansatz Compatibilty of boundary conditions and differential operator Set of admissible controls Derivation of optimality conditions (cont.) Assignment No 20: 4 Compatible formulation of parabolic equations and solution via Green s function The spacially one-dimensional case One-dimensional model problems Model with boundary control Assumptions 3.1 Some remarks of the modelling background Model with distributed control Integral representation of solution Green s function Separation of variables, Fourier s method Definition of linear operators, solution operator Assignment No 21: 5 Spacially onedimensional parabolic optimal control problems Necessary optimality conditions The boundary control cases Lemma 3.2: Representation of the adjoint solution operator Theorem 3.3: Optimality conditions for the 1D case Lemma 3.4: The variational inequality for the 1D case Corollary: Projection formula Assignment No 22: 6 Abstract functions The bang-bang principle with boundary control Theorem 3.5: The bang-bang principle Corollary 3.6: Uniqueness of optimal control Open problems 3.3 Weak solutions in W 1,0 2 (Q) 7 General Assumptions 3.7 Aim: What is a solution? Definition: The function space W 1,0 2 (Q) 4 in Zyklen No. 1-3: No in Zyklen No. 1-3: No in Zyklen No. 1-3: No. 19

12 Definition: Weak derivatives in W 1,0 2 (Q) Definition: The function space W 1,1 2 (Q) Formal derivation of a variational formulation Definition: Weak solution Theorem 3.8: Existence of a weak solution Corollary 3.9: Certain linear mappings Disadvantage of the variational formulation 3.4 Weak solutions in W(0, T) Abstract functions Definition: Abstract functions s Other important abstract functions Definition: Space of continuous abstract functions Riemann integral on C ([0, T], L 2 (Ω)) Definition: Step function Definition: Measuarable abstract function Definition: L p (a, b; X), 1 p Definition: Blochner integral : L 2 (0, T; H 1 (Ω)) Assignment No 23: 7 Parabolic optimal control problems I Abstract functions and parabolic equations New variational formulation Future road map Vector-valued distributions Aim: Derivation of a formula for partial integration for abstract functions Definition: Vector-valued distributions Definition: The Hilbert space W(0, T) Gelfand trippel Definition: Gelfand trippel Application to parabolic equations Theorem 3.10: W(0, T) C([0, T], H) Corollary Theorem 3.11: Partial integration Corallary Affiliation of weak solutions in W 1,0 2 (Q) to W(0, T) Aim: Weak solutions of parabolic equations belong to W(0, T) Theorem 3.12: Existence of weak solutions of parabolic equations in W(0, T) Theorem 3.13: Continuous dependency on data Corallary: Solution operator Results: Existence of y t, final variational formulation, 7 in Zyklen No. 1-3: No. 20

13 solution operator 3.5 Parabolic optimal control problems 4 Aim: Parabolic optimal control problems and abstract functions General assumptions Optimal instationary boundary control problems to track a prescribed final temperature distribution Associated abstract optimization problem Theorem 3.15: Existence and uniqueness of an optimal solution Optimal instationary temperature source Assignment No 24: 8 Parabolic optimal control problems II Necessary optimality conditions 21 Road map Auxiliary theorem for adjoint operator Lemma 3.16: Existence and uniqueness of solution of adjoint equation Theorem 3.17: Auxiliary theorem Optimal instationary boundary temperature Theorem 3.18: Necessary and sufficient optimality conditions Minimum principles Optimality system Generalisations Parabolic equations with general elliptic operator Parabolic equations in L 2 (0, T; V ) Assignment No 25: 9 Numerical solution of parabolic optimal control problems Numerical solution techniques Gradient projection method Preliminary considerations The algorithm Transformation to finite dimensional problem Gradient projection method The algorithm Remarks 8 in Zyklen No. 1-3: No in Zyklen No. 1-3: No. 22

14 4. Optimal Control of semilinear elliptic equations A semilinear elliptic model problem Road map Overview Solutions in H 1 (Ω) Idea Definitions: monotonous, strictly monotonous, coercive, hemi-continuous and strongly monotonous operators Analoga in R Theorem 4.1: On monotonous operators Weak formulation of semi-linear elliptic problems General assumptions 4.2 General assumptions 4.3 (to be released lateron) Definition: Weak solution Theorem 4.4: Existence of H 1 (Ω)-weak solutions Continuous Solutions Road map Theorem 4.5: Existence of bounded H 1 (Ω)-weak solutions Lemma 4.6: Continuous weak solution for a linear model problem Theorem 4.7: Continuous weak solution of the semi-linear model problem (1st version) Theorem 4.8: Continuous weak solution of the semi-linear model problem (2nd version) Assignment No 15: 10 Elliptic optimal control problems with data in the dual space Nemyzki operators Continuity of Nemyzki operators Notation Definition: Nemyzki-operators or superposition operators s Carathéodory condition Boundness condition Local Lipschitz condition s Lemma 4.9: Continuity of Nemyzki operators Remark: Lipschitz condition for Nemyzki operators 10 in Zyklen No. 1-3: No. 18

15 4.2.2 Differentiability of Nemyzki operators Definition: k-th order boundness condition, k-th order local Lipschitz condition 1st order derivatives of Nemyzki operators in L (E) : sin(y)-operator is not differentiable from L p (E) to L p (E) Discussion of differentiablity of sin(y)-operator in other function spaces Lemma 4.10: Fréchet-differentiability of Nemyzki operators in L (E) Corollary: Smoother generating functions depending on y only Continuously differentiable Nemyzki operators Definition: Continuous Fréchet differentiability of Nemyzki operators Lemma 4.11: On continuously differentiable Nemyzki operators : Φ(y) = y n Derivatives in other L p -spaces Summary on continuity Summary on differentiablity : Φ(y) = y k Assignment No 16: 11 Frechet differentiabilty of Nemyzki operators Existence of optimal solutions General assumptions General Assumption 4.12 s of functions satisfying these general assumptions Distributed control : distributed control Definitions: optimal vs. locally optimal Properties of the objective functions Theorem 4.13: Existence of optimal control Proof of Theorem 4.13 via the Theorem of Rellich Remarks on the proof Assignment No General functionals The control-state operator Distributed control Theorem 4.14: Lipschitz continuity of the solution operator Theorem 4.15: Differentiability of the solution operator Corollary Boundary control Results 11 in Zyklen No. 1-3: No in Zyklen No. 1-3: No. 20

16 4.5 Necessary optimality conditions Distributed control Aim Variational inequality Linearized boundary value problem Definition of the adjoint state Lemma 4.16: Auxiliary Lemma Corollary: Variational inequality Theorem 4.17: Necessary condition Corollary 4.18: Minimum principle : Superconductivity Boundary control Control-state operator Variational inequality Adjoint state Theorem 4.19: Necessary conditions Assignment No 18: 13 Semilinear elliptic optimal control problems Application of the formal Lagrange principle 2 General problem Definition of the Lagrange function Necessary conditions Assignment No 19: 14 Formal Lagrange Principle and Maximum Prinziple Derivatives of second order 8 Definition: twice Fréchet differentiable Twice continuously Fréchet differentiable Computation of norms Computation of F (u) : Nemyzki operator Theorem 4.20: Second derivatives of Nemyzki operators in L Counterexample 4.8 Optimality conditions of second order Introduction Two-norm discrepancy Theorem 4.21: Quadratic growth condition of the functional Counterexample Two norm discrepancy Resumee 13 in Zyklen No. 1-3: No in Zyklen No. 1-3: No. 16

17 4.8.2 Distributed control Theorem 4.22: Second continuous Fréchet derivative of the solution operator Second order Fréchet derivative of the functional Definition: Lagrange function Second order Fréchet derivative of Lagrange function Derivation of second order optimality conditions Auxiliary Lemma 4.23: Estimate for the second derivative of the functional Second order optimality conditions Definition: Critical cone Theorem 4.24: Second order necessary condition Lemma 4.25: Second order necessary condition (Lagrange version) Second order sufficient optimality conditions Another critical cone Remarks on the critical cones, strongly active constraints Theorem 4.26: Second order sufficient condition Lemma 4.27: Second order sufficient condition (Lagrange version) Boundary control Lagrange function Second order optimality conditions Second derivative of thelagrange function 4.9 Numerical methods Gradient projection method Some Remarks on the drawback of nonlinearities Basic idea of SQP method Newton s Method vs linear-quadratic optimization in finite dimensions The unconstrained case The constrained case Some Remarks to and Results for the SQP method SQP method for elliptic problems (distributed control) Assumptions Linearization (unconstrained case) Constrained case Final remarks on SQP methods

18 5. Optimal Control of semilinear parabolic equations A semilinear parabolic model problem 5 A general model problem Assumptions 5.1: Measurability, monotonicity Remark on the unboundness of the nonlinear functions Assumptions 5.2: Uniform boundness, global Lipschitz condition Definition: Weak solution Lemma 5.3: Eistence and uniqueness of solutions Assumptions 5.3: Local boundness, Local Lipschitz condition Higher regularity of data Theorem 5.5: Existence and uniqueness of continuous solutions 5.2 General Assumptions 2 C 1,1 -domains, differentiablity, boundness and Lipschitz conditions up to the order of two 5.3 Existence of optimal controls 8 for distributed and boundary controls Definition: optimal, locally optimal Definition: convexity of objective functions Theorem 5.7: Existence of optimal control 5.4 Control-state operator 10 Road map: Continuity and differentiability of the control-state operator Theorem 5.8: Lipschitz continuity of the control-state operator Theorem 5.9: Fréchet differentiablity of the control-state operator Remark on certain nonlinear problems concerning L r spaces : Distributed control : Boundary control 5.5 Necessary optimality conditions Distributed control Variational inequality Computation of f Lemma 5.10: Elimination the state y(u) for all u U ad Corollary: Computation of the gradient of the objective functionals Theorem 5.11: Necessary condition Corollary: Minimum principles : Supraconductivity (Assignement No. 25)

19 5.5.2 Boundary control Computation of f and variational inequality Theorem 5.12: Necessary condition : Heat equation with Stefan-Bolzmann boundary condition (Assignement No. 25) The general case Theorem 5.13: Necessary condition Assignment No 26: 15 Fréchet Derivatives and second-order optimality conditions Second-order optimality conditions Second-order derivatives Theorem 5.14: Second-order differentiablity of the control-state operator Theorem 5.15: Second-order derivative of the control-state operator Distributed control Variational inequality Formulation of second-order necessary conditions Definition: Strongly active constraints Definition: τ-critical cone Theorem 5.16: Second-order sufficient conditions Assignment No 27: 16 SQP methods Boundary control Theorem 5.17: Second-order sufficient conditions 5.7 Test examples 8 Road map: Method of constructing test examples Problem with control constraints (Assignement No. 27) Lemma 5.18: Global optimality of the solution (Assignement No. 27) Assignment No 28: in Zyklen No. 1-3: No in Zyklen No. 1-3: No in Zyklen No. 1-3: No. 25

20 5.8 Numerical methods 6 General problem Gradient method Algorithm: Gradient-projection method SQP method Algorithm: SQP method Remarks: Convergence, modifications, outlook 5.9 Instationary Navier-Stokes Equations 8 Some applications Simplified model problem Divergence free vector spaces Notations Definition: Weak solution of the Navier-Stokes Equations Theorem 5.19: Existence and uniqueness (Teman, 1979) Optimality conditions: First-order necessary conditions Adjoint Navier-Stokes Equations Variational inequality Remarks 6. Pontryagin s maximum principles Semilinear elliptic equations Hamilton s function Formal description Definition: Hamilton functions Necessary conditions Maximum principle The maximum principle Remarks 6.2 Semilinear parabolic equations 2 Definition: Hamilton functions Definition: Maximum principle Remarks

21 7. Optimization problems in Banach spaces Karush-Kuhn-Tucker conditions Convex problems The Lagrangmultiplier rule Definition: Convex cone Definition: Dual cone s Optimization problem in Banach spaces Definitions: Lagrange function, saddle point, Lagrange multiplier Definition: partially ordered Theorem 7.1: Necessary conditions for the convex case Slater condition : The interior of all non-negative functions in L p (p < ) is empty Remark Theorem 7.2: Necessary conditions under additional differentiablity assumptions s: One- and two-sided box constraints Remarks Non-convex problems Definition: local solution Definition: associated Lagrange multplier Definition: regularity assumption of Kurzyusz and Zowe Theorem 7.3: Existence of a Lagrange multiplier Remarks Discussion of the regularity assumption of Kurzyusz and Zowe s: One- and two-sided box constraints Remark on sufficient conditions A semilinear elliptic problem Problem formulation Necessary conditions Remarks 7.2 State constrained problems Convex problems Formulation as optimization problem in Banach spaces Lagrange function Regularity assumptions and existence of a multiplier Adjoint equations Remarks on some difficulties involved with them Theorem 7.4: Necessary conditions for the convex case Box constraints

22 Mimimization of point functionals Best approximation in the maximum norm Non-convex problems Formulation as optimization problem in Banach spaces The two-step strategy Theorem 7.5 and 7.6: Necessary conditions for the non-convex case Theorem 7.7: Necessary conditions as Karush-Kuhn-Tucker system Assignment No 29: in Zyklen No. 1-3: No. 26