138 F. Rampazzo – C. Sartori
3. Reparameterizations and Bellman equations
The contents of this section thoroughly relies on the results of [8]. Let us embed the unperturbed and the perturbed problems in a class of extended problems which have the advantage of involv-
ing only bounded controls. There is a reparameterization argument behind this embedding which allows one to transform a L
β
constraint implicitly imposed by the coercivity assumptions into a L
∞
constraint. Let us introduce the extended fields
f t, x, w , w
. =
f t, x,
w w
· w
β
if w 6= 0
f
∞
t, x, v, w if w
= 0 and α = β and
lt, x, w , w
. =
l t, x,
w w
· w
β
if w 6= 0
l
∞
t, x, v, w if w
= 0 and α = β . Similarly, for every n we define the extended fields f
n
and l
n
of f
n
and l
n
, respectively. Hy-
potheses A
1
-A
5
imply the following properties for the maps f
n
, l
n
, f , and l. P
ROPOSITION
1. i The functions f
n
, l
n
, f , and l are continuous on [0, T ] ×
k
× [0,
+∞[×
m
and for every compact Q ⊂
k
we have
A
e
1
| f t
1
, x
1
, w , w
− f t
2
, x
2
, w , w
| ≤ w
α
+ |w|
α
w
β−α
L |x
1
− x
2
| + ρ
f
|t
1
− t
2
| , | f
n
t
1
, x
1
, w , w
− f
n
t
2
, x
2
, w , w
| ≤ w
α
n
+ |w|
α
n
w
β−α
n
L |x
1
− x
2
| + ρ
f
|t
1
− t
2
| and
A
e
3
|lt
1
, x
1
, w , w
− lt
2
, x
2
, w , w
| ≤ w
β
+ |w|
β
ρ
l
|t
1
, x
1
− t
2
, x
2
| , |l
n
t
1
, x
1
, w , w
− l
n
t
2
, x
2
, w , w
| ≤ w
β
+ |w|
β
ρ
l
|t
1
, x
1
− t
2
, x
2
| ∀t
1
, x
1
, w , w,
t
2
, x
2
, w , w
∈ [0, T ] ×
k
× [0, +∞[×
m
, where α, α
n
, β, L, ρ
f
, and ρ
l
are the same as in assumptions A
1
and A
3
. Moreover,
A
e
2
| f t, x, w , w
| ≤ w
α
+ |w|
α
w
β−α
M
1
1 + |x| + M
2
, | f
n
t, x, w , w
| ≤ w
α
n
+ |w|
α
n
w
β−α
n
M
1
1 + |x| + M
2
and A
e
4
lt, x, w , w
≥ 3 |w|
β
− 3
1
|w |
β
, l
n
t, x, w , w
≥ 3 |w|
β
− 3
1
|w |
β
∀t, x, w , w
∈ [0, T ] ×
k
× [0, +∞[×
m
, where M
1
, M
2
, 3 and 3
1
are the same
as in A
2
and A
4
.
On perturbation 139
ii Positive homogeneity in w
, w . The map f , l, f
n
, and l
n
are positively homogeneous of degree β in w
, w , that is,
f t, x, r w ,
r w = r
β
f t, x, w , w,
f
n
t, x, r w ,
r w = r
β
f
n
t, x, w , w,
lt, x, r w ,
r w = r
β
lt, x, w , w
l
n
t, x, r w ,
r w = r
β
l
n
t, x, w , w
∀r 0, ∀t, x, w , w
∈ [0, T ] ×
k
×]0, +∞[×
m
. For every
¯t ∈ [0, T ] let us introduce the following sets of space-time controls Ŵ
¯t .
= w
, w ∈ [0, 1], [0, +∞ ×
m
such that ¯t +
Z
1
w
β
s ds = T
and Ŵ
+
¯t .
= w
, w ∈ Ŵ¯t such that w
0 a.e. where [0, 1], [0,
+∞×
m
is the set of L
∞
, Borel maps, which take values in [0, +∞[×
m
. If α β [resp.α
= β], for every ¯t, ¯x ∈ [0, T ] ×
k
and every w , w
∈ Ŵ
+
¯t [resp. w
, w ∈ Ŵ¯t], let us denote by t, y
¯ t, ¯x
[w , w
] · the solution of the extended Cauchy
problem E
e
t
′
s = w
β
s y
′
s = f t s, ys, w
s, ws t 0, y0
= ¯t, ¯x , where the parameter s belongs to the interval [0, 1] and the prime denotes differentiation with
respect to s. When the initial conditions are meant by the context we shall write t, y[w , w
] ·
instead of t, y
¯ t, ¯x
[w , w
] ·. Let us consider the following extended cost functional
J
e
¯t, ¯x, w , w
. =
Z
1
l t, y[w , w
], w , w
s ds + gy[w
, w ]1
and the corresponding extended value function V
e
: [0, T ] ×
k
→ V
e
¯t, ¯x .
= inf
w ,w∈Ŵ¯
t
J
e
¯t, ¯x, w , w .
Similarly, for every n ∈ , for every ¯t, ¯x ∈ [0, T ] ×
k
and every w , w
∈ Ŵ¯t let us introduce the system
E
e
n
t
′
s = w
β
s y
′
s = f
n
t s, ys, w s, ws
s ∈ [0, 1]
t 0, y0 = ¯t, ¯x ,
and let us denote its solution by t, y
n ¯
t, ¯x
[w , w
] ·. Let us introduce the cost functionals
J
e
n
¯t, ¯x, w , w
. =
Z
1
l
n
t, y
n ¯
t, ¯x
, w , w
s ds + g
n
y
n
[w , w
]1
140 F. Rampazzo – C. Sartori
and the corresponding value functions V
e
n
: [0, T ] ×
k
→ V
e
n
¯t, ¯x .
= inf
w ,w∈Ŵ¯
t
J
e
n
¯t, ¯x, w , w .
Next theorem establishes the coincidence of the value functions of the original problems with those of the extended problems.
T
HEOREM
2. Assume A
Kategori