Exercise 1. Prove:

V4E2 - Numerical Simulation

Sommersemester 2017 Prof. Dr. J. Garcke

G. Byrenheid

Exercise sheet 4. To be handed in on Thursday, 18.05.2017.

Exercise 1. Prove:

(i) Let η(·) be a nonnegative, absolutely continuous function on [0, T ], which satisfies for a.e. 0 ≤ t ≤ T the differential inequality

η ⁰ (t) ≤ ω(t)η(t) + ψ(t)

where ω(t) and ψ(t) are nonnegative, integrable functions on [0, T ]. Then η(t) ≤ e ^R

^ω(s)ds h

η(0) + Z t

0

ψ(s)ds i

for all 0 ≤ t ≤ T .

(ii) Let φ(·) be a nonnegative, integrable function on [0, T ] which satisfies for a.e. 0 ≤ t ≤ T the integral inequality

φ(t) ≤ C 2 + Z t

0

C 1 φ(s)ds

for constants C 1 , C 2 > 0. Then

φ(t) ≤ C 2 (1 + C 1 te ^C

^t )

for a.e. 0 ≤ t ≤ T . Hints: (i): consider _ds ^d (η(s)e ^{− R}

^ω(r)dr ) (ii): Use (i) to prove (ii).

(6 Punkte) Exercise 2. Prove: Assuming

(i) f : R ⁿ × A → R ⁿ is continuous such that

|f (x, α)| ≤ C, |f (x, α) − f (y, α)| ≤ C|x − y|, ∀x, y ∈ R ⁿ , α ∈ A (ii) ψ : R ⁿ → R

|ψ(x)| ≤ C, |ψ(x) − ψ(y)| ≤ C|x − y|, ∀x, y ∈ R ⁿ (iii) ` : R ⁿ × A → R

|`(x, α)| ≤ C, |`(x, α) − `(y, α)| ≤ C|x − y|, ∀x, y ∈ R ⁿ , α ∈ A.

Then

V (x, t) := inf

α∈A J _x,t (α) = inf

α∈A

Z T t

`(y _x,t (s), α(s))dt + ψ(y _x,t (T )) (1) is bounded and Lipschitz continuous, i.e.

|V (y, s)| ≤ C ⁰ , |V (y, s) − V (y ⁰ , s ⁰ )| ≤ C ⁰ (|s − s ⁰ | + |y − y ⁰ |), ∀x, y ∈ R ^d , 0 ≤ s, s ⁰ ≤ T (As usual: y _x,t (s) is the unique solution of the state dynamics generated by x, t and f ) Hints:

1

• consider 2 trajectories z(s) := y x,t (s, α ε ), z ⁰ (s) := y x

,t

(s, α ε ) with different initial data and where α ε is an almost optimal control for the initial data (x, t),

• prove |z ⁰ (t) − z(t)| ≤ C|t − t ⁰ | + |x ⁰ − x|,

• prove |z ⁰ (s) − z(s)| ≤ e ^CT (C|t − t ⁰ | + |x ⁰ − x|) using Exercise 1),

• estimate J _x,t (α _ε ) − J _x

_,t

(α _ε ).

(6 Punkte) Exercise 3. Given the initial data y(t 0 ) = x 0 . We define the function

h(t) :=

Z t t

`(y _x

_,t

(s), α(s))ds + V (y _x

_,t

(t), t).

where V is the value function known from (1). Prove:

(i) h is nondecreasing for any control α,

(ii) h is constant if and only if the control α is optimal.

(4 Punkte)

2