V4E2 - Numerical Simulation

(1)

V4E2 - Numerical Simulation

Sommersemester 2017 Prof. Dr. J. Garcke

G. Byrenheid

Exercise sheet 9. To be handed in on Tuesday, 04.07.2017.

The infinite horizon problem

Let y x denote the unique solution of the Cauchy problem ( y(s) = ˙ f (y(s), α(s))

y(0) = x.

We aim to minimize the cost

J(x, α) :=

Z ∞ 0

`(y x (t), α(t))e ^−λt dt.

For that purpose we define the value function as v(x) := inf

α∈A J (x, α).

Prerequisites

Let A ⊂ R ^M compact.

(A 0 )

( A is a topological space,

f : R ^N × A → R ^N is continuous, (A 1 ) f is bounded on B(0, R) × A for all R > 0,

(A 2 ) there is a modulus ω f such that

|f (y, a) − f (x, a)| ≤ ω _f (|x − y|, R), for all x, y ∈ B (0, R) and R > 0.

(A 3 )

(f (x, a) − f(y, a)) · (x − y) ≤ L|x − y| ² for all x, y ∈ R ^N , a ∈ A.

(A 4 ) • ` is continuous,

• there are modulus ω _` and a constant M such that

|`(x, a) − `(y, a)| ≤ ω _` (|x − y|) and

|`(x, a)| ≤ M, for all x, y ∈ R ^N and a ∈ A,

• λ > 0

1

(2)

Exercises

Exercise 1. Prove: Assume (A 0 ), (A 1 ), (A 3 ), and (A 4 ). Then v ∈ BU C(R ^N ). If moreover ω _` (r) = L _` r (i.e., ` is Lipschitz in y, uniformly in a), then v is H¨ older continuous with the following exponent γ:

γ =



 

 

1 if λ > L

any γ < 1 if λ = L

γ

L if λ < L

Hint: Under the assumptions from above a basic property of y _x is

|y _x (t, α) − y z (t, α)| ≤ e ^Lt |x − z|

for all α ∈ A, and t > 0. L denotes the constant known from (A ₃ ).

(6 Punkte) Feedback maps

Definition 1. A control law or presynthesis on a set Ω ⊆ R ^N is a map A : Ω → A, that is, it associates with each point x ∈ Ω a control function A (x) =: a x . It is optimall on Ω if the cost associated with it, that is, J _A (x) := J (x, α _x ), satisfies

J _A (x) = min

α∈A J (x, α) = v(x) for all x ∈ Ω.

The most important examples of control laws are generated by feedback maps Ψ : Ω → A, provided the feedback is admissible in the following sense.

Definition 2. A feedback map on a set Ω ⊆ R ^N , Ψ : Ω → A, is admissible if for all x ∈ Ω there exists a unique solution y x (·, Ψ) on [0, +∞[ of

( (y) = ˙ f (y, Ψ(y)) y(0) = x

such that t → Ψ(y x (t, Ψ)) is measurable and y x (t, Ψ) ∈ Ω for all t ≥ 0.

It is natural to associate the following control law with an admissible feedback map α _x (·) := Ψ(y _x (·, Ψ)) ∈ A;

α x in this case is called a closed-loop control.

Exercise 2. Denote by F the set of admissible feedback maps on R ^N , and set, for Ψ ∈ F , J F (x, Ψ) :=

Z ∞ 0

e ^−λt `(y _x (t, Ψ), Ψ(y _x (t, Ψ)))dt v F (x) = inf

Ψ∈F J F (x, Ψ).

(i) Prove that v F = v. [Hint: one inequality is trivial, the other is easily obtained by adding time as a state variable.]

(ii) Prove directly (without using (i)) that v F is continuous and satisfies the Dynamic Pro- gramming Principle.

(6 Punkte) Treating the convex HJ equation in multiple space dimensions

We consider Hamilton-Jacobi equations in multiple space dimensions

u t + H(u x

1

, . . . , u x

d

) = 0, R ^d × [0, T ] (1) Exercise 3. Generate the upwind scheme for (1) in case d = 2. Prove monotonicity and consistency for that scheme. (Adapt the univariate proofs).

(6 Punkte)

2

V4E2 - Numerical Simulation

V4E2 - Numerical Simulation

Sommersemester 2017 Prof. Dr. J. Garcke

G. Byrenheid

Exercise sheet 9. To be handed in on Tuesday, 04.07.2017.

The infinite horizon problem

Let y x denote the unique solution of the Cauchy problem ( y(s) = ˙ f (y(s), α(s))

y(0) = x.

We aim to minimize the cost

J(x, α) :=

Z ∞ 0

`(y x (t), α(t))e −λt dt.

For that purpose we define the value function as v(x) := inf

α∈A J (x, α).

Prerequisites

Let A ⊂ R M compact.

(A 0 )

( A is a topological space,

f : R N × A → R N is continuous, (A 1 ) f is bounded on B(0, R) × A for all R > 0,

(A 2 ) there is a modulus ω f such that

|f (y, a) − f (x, a)| ≤ ω f (|x − y|, R), for all x, y ∈ B (0, R) and R > 0.

(A 3 )

(f (x, a) − f(y, a)) · (x − y) ≤ L|x − y| 2 for all x, y ∈ R N , a ∈ A.

(A 4 ) • ` is continuous,

• there are modulus ω ` and a constant M such that

|`(x, a) − `(y, a)| ≤ ω ` (|x − y|) and

|`(x, a)| ≤ M, for all x, y ∈ R N and a ∈ A,

• λ > 0

1

Exercises

Exercise 1. Prove: Assume (A 0 ), (A 1 ), (A 3 ), and (A 4 ). Then v ∈ BU C(R N ). If moreover ω ` (r) = L ` r (i.e., ` is Lipschitz in y, uniformly in a), then v is H¨ older continuous with the following exponent γ:

γ =



 

 

1 if λ > L

any γ < 1 if λ = L

γ

L if λ < L

Hint: Under the assumptions from above a basic property of y x is

|y x (t, α) − y z (t, α)| ≤ e Lt |x − z|

for all α ∈ A, and t > 0. L denotes the constant known from (A 3 ).

(6 Punkte) Feedback maps

Definition 1. A control law or presynthesis on a set Ω ⊆ R N is a map A : Ω → A, that is, it associates with each point x ∈ Ω a control function A (x) =: a x . It is optimall on Ω if the cost associated with it, that is, J A (x) := J (x, α x ), satisfies

J A (x) = min

α∈A J (x, α) = v(x) for all x ∈ Ω.

The most important examples of control laws are generated by feedback maps Ψ : Ω → A, provided the feedback is admissible in the following sense.

Definition 2. A feedback map on a set Ω ⊆ R N , Ψ : Ω → A, is admissible if for all x ∈ Ω there exists a unique solution y x (·, Ψ) on [0, +∞[ of

( (y) = ˙ f (y, Ψ(y)) y(0) = x

such that t → Ψ(y x (t, Ψ)) is measurable and y x (t, Ψ) ∈ Ω for all t ≥ 0.

It is natural to associate the following control law with an admissible feedback map α x (·) := Ψ(y x (·, Ψ)) ∈ A;

α x in this case is called a closed-loop control.

Exercise 2. Denote by F the set of admissible feedback maps on R N , and set, for Ψ ∈ F , J F (x, Ψ) :=

Z ∞ 0

e −λt `(y x (t, Ψ), Ψ(y x (t, Ψ)))dt v F (x) = inf

Ψ∈F J F (x, Ψ).

(i) Prove that v F = v. [Hint: one inequality is trivial, the other is easily obtained by adding time as a state variable.]

(ii) Prove directly (without using (i)) that v F is continuous and satisfies the Dynamic Pro- gramming Principle.

(6 Punkte) Treating the convex HJ equation in multiple space dimensions

We consider Hamilton-Jacobi equations in multiple space dimensions

u t + H(u x

, . . . , u x

) = 0, R d × [0, T ] (1) Exercise 3. Generate the upwind scheme for (1) in case d = 2. Prove monotonicity and consistency for that scheme. (Adapt the univariate proofs).

(6 Punkte)

2

`(y x (t), α(t))e ^−λt dt.

Let A ⊂ R ^M compact.

f : R ^N × A → R ^N is continuous, (A 1 ) f is bounded on B(0, R) × A for all R > 0,

|f (y, a) − f (x, a)| ≤ ω _f (|x − y|, R), for all x, y ∈ B (0, R) and R > 0.

(f (x, a) − f(y, a)) · (x − y) ≤ L|x − y| ² for all x, y ∈ R ^N , a ∈ A.

• there are modulus ω _` and a constant M such that

|`(x, a) − `(y, a)| ≤ ω _` (|x − y|) and

|`(x, a)| ≤ M, for all x, y ∈ R ^N and a ∈ A,

Exercise 1. Prove: Assume (A 0 ), (A 1 ), (A 3 ), and (A 4 ). Then v ∈ BU C(R ^N ). If moreover ω _` (r) = L _` r (i.e., ` is Lipschitz in y, uniformly in a), then v is H¨ older continuous with the following exponent γ:

Hint: Under the assumptions from above a basic property of y _x is

|y _x (t, α) − y z (t, α)| ≤ e ^Lt |x − z|

for all α ∈ A, and t > 0. L denotes the constant known from (A ₃ ).

Definition 1. A control law or presynthesis on a set Ω ⊆ R ^N is a map A : Ω → A, that is, it associates with each point x ∈ Ω a control function A (x) =: a x . It is optimall on Ω if the cost associated with it, that is, J _A (x) := J (x, α _x ), satisfies

J _A (x) = min

Definition 2. A feedback map on a set Ω ⊆ R ^N , Ψ : Ω → A, is admissible if for all x ∈ Ω there exists a unique solution y x (·, Ψ) on [0, +∞[ of

It is natural to associate the following control law with an admissible feedback map α _x (·) := Ψ(y _x (·, Ψ)) ∈ A;

Exercise 2. Denote by F the set of admissible feedback maps on R ^N , and set, for Ψ ∈ F , J F (x, Ψ) :=

e ^−λt `(y _x (t, Ψ), Ψ(y _x (t, Ψ)))dt v F (x) = inf

) = 0, R ^d × [0, T ] (1) Exercise 3. Generate the upwind scheme for (1) in case d = 2. Prove monotonicity and consistency for that scheme. (Adapt the univariate proofs).