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1. Exercise

1. Assume f (x) = Ax · x + b · x A is a positive-definite matrix, b any vector in R

N

. Show that f is coercive, that is

lim

|x|→+∞

f (x) = +∞

2. Write the matrix associated to the quadratic form 3λ

21

+ 2λ

1

λ

2

+ 2λ

3

λ

2

− λ

22

− λ

23

3. The Dynamic programming principle DPP

u(x) = inf

α

 Z

t

0

f (X

x

(s), α(s))e

−λs

ds + u(X

x

(t))e

−λt

, for all real x and for all positive t.

The dynamic programming principle has been shown taking u ∈ C

1

(R

n

).

Show that u continuous in R

n

satisfying DPP, verifies the property (a). For any ϕ ∈ C

1

(R

n

) such that u − ϕ has a local maximum in x, then

λu(x) + max

a

{−Dϕ(x) · b(x, a) − f (x, a)} ≤ 0.

Hint: Use u(x) − ϕ(x) ≥ u(X

x

(s)) − ϕ(X

x

(s)) for s small for s small ) to get u(x) − u(X

x

(s)) ≥ ϕ(x) − ϕ(X

x

(s)) for s small... )

(b). For any ϕ ∈ C

1

(R

n

) such that u − ϕ has a local minimum in x, then λu(x) + max

a

{−Dϕ(x) · b(x, a) − f (x, a)} ≥ 0.

1

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