Theorem 2.11: Existence and uniqueness of an optimal solution of the model problem (1st version

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0. Historical Remarks 4

• The pioneers of optimal control theory

• Optimal control of ordinary differential equations

• Optimal control of partial differential equations

1. Introduction 35

1.1 Examples for convex problems 7

1.1.1 Optimal stationary heating

• Optimal boundary temperature

• Model problem

• First discussion

• Optimal temperature source

• Model problem

1.1.2 Optimal instationary heating

• Model problem 1.1.3 Optimal vibrating

• Model problem

1.2 Examples for non-convex problems 3

1.2.1 Problems with semilinear elliptic equations

• Heating by thermal radiation

• Ginzburg-Landau equation in supraconductivity

• Control of stationary currents

1.2.2 Problems with semilinear parabolic equations

• Examples

• Instationary fluid flow control

1.3 Classification of 2nd order partial differential equations 7 1.3.1 Formal classification

• Definitions: Discriminant of a partial differential equation, elliptic, parabolic, hyperbolic

• Standard examples and others

1.3.2 Non-formal classification of 2nd order partial differential equations

• Model problem

• Necessity of a classification

• Definitions: elliptic, parabolic, hyperbolic

• Examples and characteristic curves

• Suitable initial and boundary conditions

Assignment No 1: Classification of PDEs 12

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1.4 Basic concepts of optimization in the finite dimensional case 18 1.4.1 Finite dimensional problems of optimal control

• Model problem

• Reformulation

1.4.2 Existence of optimal controls

• Definition: optimal control, optimal state

• Theorem 1.1: Existence theorem 1.4.3 First order necessary conditions

• Idea

• Notations

• Theorem 1.2: Variational inequality

• Example 1.4.4 Adjoint state

• Idea

• Definition: adjoint equation

• Example

• 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)

• Example 1.4.5 Lagrange function

• Aim

• Definition: Lagrange function

• Reformulation of the optimality system 1.4.6 Discussion of the variational inequality

• Discussion

1.4.7 Formulation as Karush-Kuhn-Tucker system

• Eliminating inequality constraints

• Theorem 1.4: Existence of Lagrange multipliers, optimality system

• Definitions: (strongly) active inequality constraints²

• Critical cone³

• Theorem 1.5: Sufficient condition³

• Outlook on pde constrained optimization

2presented in Chapter 2.11.4

3not presented in course

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2. Linear quadratic elliptic control problems 119

2.1 Normed vector spaces 3

• Normed linear spaces

• Convergence of sequences, Cauchy sequences

• Banach spaces

• Pre-Hilbert spaces

• Hilbert spaces

2.2 Sobolev spaces 10

2.2.1 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 2.2.2 Regular domains

• Definition: domains of class C^k,¹

• Introduction of a Lebesgue measure on the boundary of a domain 2.2.3 Weak derivatives and Sobolev spaces

• Formula of partial integration

• Definition: locally integrable

• Definition: weak derivative

• Example

• 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 11

2.3 Weak solution of elliptic equations 6

2.3.1 Poisson equation

• Model problem

• 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

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2.3.2 Boundary conditions of third kind

• See exercises

2.3.3 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) 7

2.4 Linear mappings 15

2.4.1 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 8

2.4.2 Weak convergence

• Definitions: weakly convergent, weak limit point

• Some results on weakly convergent sequences

• Example 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

• Examples 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

• Example: norm

Assignment No 7: Weak convergence and related terms 8

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2.5 Existence of optimal controls 15 2.5.1 Optimal stationary temperature control problem

with a distributed control

• Model problem: 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)

• Model problem: optimal stationary temperature source with prescribed ambient temperature

• Results on existence and uniqueness

2.5.2 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 2.5.3 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 8

2.6.1 Gˆateaux derivative

• Definitions: directional derivative, first variation, Gˆateaux derivative

• Example: The Gˆateaux derivative need not be linear

• Examples: among others: squared norm 2.6.2 Fr´echet derivative

• Definitions: Fr´echet differentiable, Fr´echet derivative

• Examples: among others: Fr´echet derivative of linear operators

• Theorem 2.15: Fr´echet differentiability implies Gˆateaux differentiablity

• Remark and Example: Counterexample that the reverse does not hold

• Chain rule

• Example

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Assignment No 8: Gˆateaux and Fr´echet differentiability 6 Assignment No 9: Calculus of variations (add on) 14

2.7 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

• Example: an integral operator

2.8 Necessary first order optimality conditions 24

2.8.1 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

2.8.2 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

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• The gradient of the objective function

• Lemma 2.27: On the gradient of the objective function

• Rewriting of the variational inequality using the gradient

2.8.3 Stationary temperature source and boundary conditions of third kind

• Problem statement

• Adjoint equation, optimality condition 2.8.4 Optimal stationary temperature control problem

with a boundary control

• 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 andλ >0 2.8.5 A linear optimal control problem

• Optimality conditions

Assignment No 10: Optimality conditions 10

2.9 Construction of test problems 5

2.9.1 Bang-bang control

• Problem statement with free parameter functions

• Construction of an analytical solution

2.9.2 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

• Example: Optimal stationary boundary control

• Discussion of mathematical inaccuracies

• Lagrange’s optimization problem

• Discussion of the formal Lagrange principle

Assignment No 11: The formal Lagrange principle 15

2.11 Numerical methods 19

2.11.1 The Conditional gradient method

• Formulation of the conditional gradient method in Hilbert spaces

• Algorithm

• Application to elliptic problems

• Discussion

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2.11.2 Transcription method

• Transformation of the infinite dimensional problem into a finite dimensional problem

• Disussion

2.11.3 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 2.11.4 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 I² 16

Assignment No 14: Novel problems II³ 24

1new in WS 2011/12

2new in WS 2011/12

3new in WS 2011/12

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3. Linear quadratic parabolic control problems 77

3.1 Introduction 6

• Elliptic versus parabolic problems

• Model problem

• 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 4

3.2 The spacially one-dimensional case 17

3.2.1 One-dimensional model problems

• Model with boundary control

• Assumptions 3.1

• Some remarks of the modelling background

• Model with distributed control

3.2.2 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 9

3.2.3 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 6

3.2.4 The bang-bang principle

• Example with boundary control

• Theorem 3.5: The bang-bang principle

• Corollary 3.6: Uniqueness of optimal control

• Open problems

3.3 Weak solutions in W2¹^,⁰(Q) 7

• Model problem

• General Assumptions 3.7

• Aim: What is a solution?

• Definition: The function space W2¹^,⁰(Q)

4in Zyklen No. 1-3: No. 17

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• Definition: Weak derivatives in W2¹^,⁰(Q)

• Definition: The function space W2¹^,¹(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) 22

3.4.1 Abstract functions

• Definition: Abstract functions

• Examples

• Other important abstract functions

• Definition: Space of continuous abstract functions

• Example

• Riemann integral on C([0, T], L²(Ω))

• Definition: Step function

• Definition: Measuarable abstract function

• Definition: L^p(a, b;X), 1≤p≤ ∞

• Definition: Blochner integral

• Example: L²(0, T;H¹(Ω))

Assignment No 23:⁷ Parabolic optimal control problems I 8 3.4.2 Abstract functions and parabolic equations

• New variational formulation

• Future road map 3.4.3 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

3.4.4 Affiliation of weak solutions in W2¹^,⁰(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,

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solution operator

3.5 Parabolic optimal control problems 4

• Aim: Parabolic optimal control problems and abstract functions

• General assumptions 3.14

3.5.1 Optimal instationary boundary control problems to track a prescribed final temperature distribution

• Model problem

• Associated abstract optimization problem

• Theorem 3.15: Existence and uniqueness of an optimal solution 3.5.2 Optimal instationary temperature source

Assignment No 24:⁸ Parabolic optimal control problems II 4

3.6 Necessary optimality conditions 21

• Road map

3.6.1 Auxiliary theorem for adjoint operator

• Lemma 3.16: Existence and uniqueness of solution of adjoint equation

• Theorem 3.17: Auxiliary theorem

3.6.2 Optimal instationary boundary temperature

• Theorem 3.18: Necessary and sufficient optimality conditions

• Minimum principles

• Optimality system 3.6.3 Generalisations

• Parabolic equations with general elliptic operator

• Parabolic equations in L²(0, T;V^∗) Assignment No 25:⁹ Numerical solution

of parabolic optimal control problems 4

3.7 Numerical solution techniques 6

3.7.1 Gradient projection method

• Preliminary considerations

• The algorithm

3.7.2 Transformation to finite dimensional problem

• Gradient projection method

• The algorithm

• Remarks

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4. Optimal Control of semilinear elliptic equations 105

4.1 A semilinear elliptic model problem 17

4.1.1 Road map

• Example

• Overview

• Model problem 4.1.2 Solutions inH¹(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)

• Theorem 4.4: Existence of H¹(Ω)-weak solutions 4.1.3 Continuous Solutions

• Road map

• Theorem 4.5: Existence of bounded H¹(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 5

4.2 Nemyzki operators 15

4.2.1 Continuity of Nemyzki operators

• Notation

• Definition: Nemyzki-operators or superposition operators

• Examples

• Carath´eodory condition

• Boundness condition

• Local Lipschitz condition

• Examples

• Lemma 4.9: Continuity of Nemyzki operators

• Remark: Lipschitz condition for Nemyzki operators

• Example

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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)

• Example: 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: Fr´echet-differentiability of Nemyzki operators in L^∞(E)

• Corollary: Smoother generating functions depending ony only

• Continuously differentiable Nemyzki operators

• Definition: Continuous Fr´echet differentiability of Nemyzki operators

• Lemma 4.11: On continuously differentiable Nemyzki operators

• Example: Φ(y) =yⁿ 4.2.3 Derivatives in other L^p-spaces

• Summary on continuity

• Summary on differentiablity

• Example: Φ(y) =y^k

Assignment No 16:¹¹ Frechet differentiabilty of Nemyzki operators 4

4.3 Existence of optimal solutions 11

4.3.1 General assumptions

• General Assumption 4.12

• Examples of functions satisfying these general assumptions 4.3.2 Distributed control

• Model problem: 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 17¹² General functionals 3

4.4 The control-state operator 8

4.4.1 Distributed control

• Model problem

• Theorem 4.14: Lipschitz continuity of the solution operator

• Theorem 4.15: Differentiability of the solution operator

• Corollary 4.4.2 Boundary control

• Model problem

• Results

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4.5 Necessary optimality conditions 7 4.5.1 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

• Example

• Example: Superconductivity 4.5.2 Boundary control

• Model problem

• Control-state operator

• Adjoint state

• Theorem 4.19: Necessary conditions

Assignment No 18:¹³ Semilinear elliptic optimal control problems 5

4.6 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 7

4.7 Derivatives of second order 8

• Definition: twice Fr´echet differentiable

• Twice continuously Fr´echet differentiable

• Computation of norms

• Computation of F^′′(u)

• Example: Nemyzki operator

• Theorem 4.20: Second derivatives of Nemyzki operators in L^∞

• Counterexample

4.8 Optimality conditions of second order 28

4.8.1 Introduction — Two-norm discrepancy

• Theorem 4.21: Quadratic growth condition of the functional

• Counterexample

• Two norm discrepancy

• Example

• Resumee

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• Theorem 4.22: Second continuous Fr´echet derivative of the solution operator

• Second order Fr´echet derivative of the functional

• Second order Fr´echet 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) 4.8.3 Boundary control

• Model problem

• Lagrange function

• Second order optimality conditions

• Second derivative of theLagrange function

4.9.1 Gradient projection method

• Some Remarks on the drawback of nonlinearities 4.9.2 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 4.9.3 SQP method for elliptic problems

• Model problem (distributed control)

• Assumptions

• Linearization (unconstrained case)

• Constrained case

• Final remarks on SQP methods

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5. Optimal Control of semilinear parabolic equations 77

5.1 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

• 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

• C¹^,¹-domains, differentiablity, boundness and Lipschitz conditions up to the order of two

5.3 Existence of optimal controls 8

• Model problem 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: Fr´echet differentiablity of the control-state operator

• Remark on certain nonlinear problems concerning L^r spaces

• Example: Distributed control

• Example: Boundary control

5.5 Necessary optimality conditions 10

• Model problem

• 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

• Corollary: Minimum principles

• Example: Supraconductivity (Assignement No. 25)

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5.5.2 Boundary control

• Model problem

• Computation of f^′ and variational inequality

• Example: Heat equation with Stefan-Bolzmann boundary condition (Assignement No. 25)

5.5.3 The general case

• Theorem 5.13: Necessary condition Assignment No 26:¹⁵ Fr´echet Derivatives

and second-order optimality conditions 5

5.6 Second-order optimality conditions 20

5.6.1 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

• 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 5

5.6.3 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 28:¹⁷ 15

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• General problem 5.8.1 Gradient method

• Algorithm: Gradient-projection method 5.8.2 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

• Remarks

6. Pontryagin’s maximum principles 6

6.1 Semilinear elliptic equations 4

6.1.1 Hamilton’s function

• Model problem

• Formal description

• Definition: Hamilton functions

• Necessary conditions 6.1.2 Maximum principle

• The maximum principle

• Remarks

6.2 Semilinear parabolic equations 2

• Model problem

• Definition: Hamilton functions

• Definition: Maximum principle

• Remarks

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7. Optimization problems in Banach spaces 41

7.1 Karush-Kuhn-Tucker conditions 23

7.1.1 Convex problems

• The Lagrangmultiplier rule

• Definition: Convex cone

• Example

• Definition: Dual cone

• Examples

• 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

• Example: The interior of all non-negative functions in L^p (p <∞) is empty

• Remark

• Theorem 7.2: Necessary conditions under additional differentiablity assumptions

• Examples: One- and two-sided box constraints

• Remarks

7.1.2 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

• Example

• Discussion of the regularity assumption of Kurzyusz and Zowe

• Examples: One- and two-sided box constraints

• Remark on sufficient conditions 7.1.3 A semilinear elliptic problem

• Problem formulation

• Necessary conditions

• Remarks

7.2 State constrained problems 18

7.2.1 Convex problems

• Model problem

• 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

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• Mimimization of point functionals

• Best approximation in the maximum norm 7.2.1 Non-convex problems

• Model problem

• 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 29:¹⁸ 25