The goal of this exercise is to model a one-dimensional parabolic optimal control problem, discretize it and derive the corresponding Lagrange formulation.

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Scientific Computing 2

Summer term 2017 Prof. Dr. Ira Neitzel Christopher Kacwin

Sheet 10 Submission on Thursday, 20.7.2017.

Exercise 1. (optimal control)

The goal of this exercise is to model a one-dimensional parabolic optimal control problem, discretize it and derive the corresponding Lagrange formulation.

We consider a metal rod and its temperature distribution y : [0, 1] × [0, T ] −→ R

with initial condition y(·, 0) = y ⁰ . Additionally,we assume that we are able to control the heat flux of the metal rod at the end points. More precisely, we model y(x, t) to satisfy the partial differential equation

y t − y xx = f in [0, 1] × [0, T ]

−y _x (0, ·) = u _l in [0, T ] y _x (1, ·) = u _r in [0, T ]

y(·, 0) = y ⁰ in [0, 1]

with control parameters u l (t), u r (t) and additional enviromental influence f (x, t) (ma- terial conditions, additional heat source...). The goal is to influence this temperature distribution such that at time T , it will be close to the desired end state y _d . This should be balanced with respect to the energy needed to advance from y ⁰ to y d . A cost functional to this problem can be stated as

J (y, u l , u r ) = 1

2 ky(·, T ) − y d k ² _L

2

[0,1] + α 2

ku _l k ² _L

2

[0,T] + ku _r k ² _L

2

[0,T ]

which we want to minimize with respect to some constraints on u _l and u r .

As a first step, we want to do a spatial discretization of the partial differential equation.

We interpret y(x, t) = y(t)(x) = y(t) ∈ V (f likewise), where y is now a function of time mapping into a function space V , which consists of functions defined on [0, 1] (for instance C[0, 1]). The finite-dimensional subspace V _h ⊂ V with basis {φ ₁ , . . . , φ _m } is used to approximate y(t) as

y(t) ≈

m

X

i=1

y _i (t)φ i

with a time-dependent coefficient vector y(t) ∈ R ^m . a) Derive the spatially discretized weak formulation

My ⁰ (t) + Ky(t) = L(t) , t ∈ [0, T ] M y(0) = I .

Here, M ∈ R ^m×m is the mass matrix with M _ij = R

φ _i φ _j , K ∈ R ^m×m is the stiffness matrix with K ij = R

(φ i ) x (φ j ) x , L(t) ∈ R ^m is the load vector with L i (t) = R f (t)φ _i + φ _i (0)u _l (t) + φ _i (1)u _r (t), and I ∈ R ^m are the initial conditions with I _i = R y ⁰ φ _i .

1

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This is a vector-valued first order ODE with matrix coefficients. We continue with a time discretization. Introducing the time steps t _n = nT /N for n = 0, . . . , N with spacing τ = T /N we define y ⁿ = y(t n ), L ⁿ = L(t n ).

b) Using the implicit Euler scheme, derive the space-time discretized formulation (M + τ K) y ⁿ = My ⁿ⁻¹ + τ L ⁿ , n = 1, . . . , N

My ⁰ = I .

State a block-matrix formulation that expresses Y = [y ⁿ ] ^N _n=1 ∈ ( R ^m ) ^N as the solution of a linear system

AY = B .

c) State the discrete optimization problem using an appropiate discrete cost functio- nal and justify your choice in a few words. Introduce the Lagrangian formalism for this problem using discrete Lagrangian multipliers p ⁿ for n = 1, . . . , N (without restrictions for the control parameters).

d) State the KKT-conditions for the discrete optimization problem, with emphasis on the adjoint equations. Do these equations resemble a certain differential equation?

(20 points)

2

The goal of this exercise is to model a one-dimensional parabolic optimal control problem, discretize it and derive the corresponding Lagrange formulation.

Scientific Computing 2

Summer term 2017 Prof. Dr. Ira Neitzel Christopher Kacwin

Sheet 10 Submission on Thursday, 20.7.2017.

Exercise 1. (optimal control)

The goal of this exercise is to model a one-dimensional parabolic optimal control problem, discretize it and derive the corresponding Lagrange formulation.

We consider a metal rod and its temperature distribution y : [0, 1] × [0, T ] −→ R

with initial condition y(·, 0) = y 0 . Additionally,we assume that we are able to control the heat flux of the metal rod at the end points. More precisely, we model y(x, t) to satisfy the partial differential equation

y t − y xx = f in [0, 1] × [0, T ]

−y x (0, ·) = u l in [0, T ] y x (1, ·) = u r in [0, T ]

y(·, 0) = y 0 in [0, 1]

J (y, u l , u r ) = 1

2 ky(·, T ) − y d k 2 L

[0,1] + α 2

ku l k 2 L

[0,T] + ku r k 2 L

[0,T ]

which we want to minimize with respect to some constraints on u l and u r .

As a first step, we want to do a spatial discretization of the partial differential equation.

y(t) ≈

m

X

i=1

y i (t)φ i

with a time-dependent coefficient vector y(t) ∈ R m . a) Derive the spatially discretized weak formulation

My 0 (t) + Ky(t) = L(t) , t ∈ [0, T ] M y(0) = I .

Here, M ∈ R m×m is the mass matrix with M ij = R

φ i φ j , K ∈ R m×m is the stiffness matrix with K ij = R

(φ i ) x (φ j ) x , L(t) ∈ R m is the load vector with L i (t) = R f (t)φ i + φ i (0)u l (t) + φ i (1)u r (t), and I ∈ R m are the initial conditions with I i = R y 0 φ i .

1

This is a vector-valued first order ODE with matrix coefficients. We continue with a time discretization. Introducing the time steps t n = nT /N for n = 0, . . . , N with spacing τ = T /N we define y n = y(t n ), L n = L(t n ).

b) Using the implicit Euler scheme, derive the space-time discretized formulation (M + τ K) y n = My n−1 + τ L n , n = 1, . . . , N

My 0 = I .

State a block-matrix formulation that expresses Y = [y n ] N n=1 ∈ ( R m ) N as the solution of a linear system

AY = B .

c) State the discrete optimization problem using an appropiate discrete cost functio- nal and justify your choice in a few words. Introduce the Lagrangian formalism for this problem using discrete Lagrangian multipliers p n for n = 1, . . . , N (without restrictions for the control parameters).

d) State the KKT-conditions for the discrete optimization problem, with emphasis on the adjoint equations. Do these equations resemble a certain differential equation?

(20 points)

2

with initial condition y(·, 0) = y ⁰ . Additionally,we assume that we are able to control the heat flux of the metal rod at the end points. More precisely, we model y(x, t) to satisfy the partial differential equation

−y _x (0, ·) = u _l in [0, T ] y _x (1, ·) = u _r in [0, T ]

y(·, 0) = y ⁰ in [0, 1]

2 ky(·, T ) − y d k ² _L

ku _l k ² _L

[0,T] + ku _r k ² _L

which we want to minimize with respect to some constraints on u _l and u r .

y _i (t)φ i

with a time-dependent coefficient vector y(t) ∈ R ^m . a) Derive the spatially discretized weak formulation

My ⁰ (t) + Ky(t) = L(t) , t ∈ [0, T ] M y(0) = I .

Here, M ∈ R ^m×m is the mass matrix with M _ij = R

φ _i φ _j , K ∈ R ^m×m is the stiffness matrix with K ij = R

(φ i ) x (φ j ) x , L(t) ∈ R ^m is the load vector with L i (t) = R f (t)φ _i + φ _i (0)u _l (t) + φ _i (1)u _r (t), and I ∈ R ^m are the initial conditions with I _i = R y ⁰ φ _i .

This is a vector-valued first order ODE with matrix coefficients. We continue with a time discretization. Introducing the time steps t _n = nT /N for n = 0, . . . , N with spacing τ = T /N we define y ⁿ = y(t n ), L ⁿ = L(t n ).

b) Using the implicit Euler scheme, derive the space-time discretized formulation (M + τ K) y ⁿ = My ⁿ⁻¹ + τ L ⁿ , n = 1, . . . , N

My ⁰ = I .

State a block-matrix formulation that expresses Y = [y ⁿ ] ^N _n=1 ∈ ( R ^m ) ^N as the solution of a linear system

c) State the discrete optimization problem using an appropiate discrete cost functio- nal and justify your choice in a few words. Introduce the Lagrangian formalism for this problem using discrete Lagrangian multipliers p ⁿ for n = 1, . . . , N (without restrictions for the control parameters).