142 F. Rampazzo – C. Sartori
for every t, x ∈ [0, T [×B[0; R], where η
t
is a suitable modulus, and for every s, t → η
t
s is an increasing map. The same statement holds true for the maps V
e
n
, with the same η
t
. R
EMARK
1. We do not need, for our purposes, an explicit expression of η
t
, which, how- ever, can be found in [8]. Also in that paper sharper regularity results are established. Finally let
us point out that though an estimate like the second one in Theorem 5 is not available for t = T the map V
e
is continuous on {T } ×
k
, see Theorem 2.
4. Proof of the convergence theorem
Proof of Theorem 1. In view of Theorem 2 it is sufficient to show that the maps V
e
n
converge to V
e
. Observe that the assumptions 4, 5 imply | f
n
t, x, w , w −
f t, x, w , w| ≤ ǫ
nw
β
+ |w|
β
8 and
|l
n
t, x, w , w −
lt, x, w , w| ≤ ǫ
nw
β
+ |w|
β
, 9
for every t, x, w , w ∈
[0, T ] × Q × [0, ∞[×
m
and every n ∈ .
Moreover, by the coercivity condition A
e
4
and by the obvious local uniform boundedness of V
e
n
, and V
e
when the initial conditions are taken in a ball B[0, R] it is not restrictive to consider only those space time controls such that
Z
1
w s + |ws|
β
ds ≤ K
R
10 where K
R
is a suitable constant depending on R. By H¨older’s inequality we have also that Z
1
w s + |ws|
α
n
w s
β−α
n
ds ≤ T + 1K
R
+ 1 . Hence by Gronwall’s Lemma, we can assume that there exists a ball B[0, R
′
] ⊂
k
containing all the trajectories issuing from B[0, R].
Let us fix T T : by Theorem 5 the maps V
e
n
are equicontinuous and equibounded on [0, T ] × B[0, R], so we can apply Ascoli-Arzela’s Theorem to get a subsequence of V
e
n
, still denoted by V
e
n
, converging to a continuous function. Actually by taking R larger and larger, via a standard diagonal procedure we can assume that the V
e
n
converge to a continuous function : [0, T ] ×
k
→ , uniformly on compact sets of [0, T ] ×
k
. Now, for every t, x ∈ [0, T ] ×
k
, let us consider the weak limits V t, x
. =
lim sup
n→∞ s,y→t,x
s,y∈[0,T ]×
✁
k
V
e
n
s, y
and V t, x
. =
lim inf
n→∞ s,y→t,x
s,y∈[0,T ]×
✁
k
V
e
n
s, y .
On perturbation 143
Our goal is to apply a method see [2] based on the application of the comparison theorem see Theorem 3 to these weak limits. Let us observe that both V and V coincide with
on the boundary {0} ×
k
: in particular they are continuous on {0} ×
k
. Since the Hamiltonians H
e
n
converge to H
e
uniformly on compact subsets of [0, T ]×
k
× ×
k
, standard arguments imply that V is a upper semicontinuous viscosity subsolution of H J
e
in [0, T ×
k
, while V is a lower semicontinuous viscosity supersolution of H J
e
in [0, T ×
k
. Hence the convergence result is proven as soon as one shows that V ≤ V in [0, T ] ×
k
. For this purpose it is sufficient to show that V and V verify the hypotheses of Theorem 3. Actually the only hypothesis which
is left to be verified is the one concerning the boundary subset {T } ×
k
. We claim that lim
n→∞ s,y→T ,x
s,y∈[0,T ]×
✁
k
V
e
n
s, y = V
e
T, x 11
which implies V T, · = V T, · = V
e
T, ·. In particular the maps V T, · and V T, · turn out to be continuous, so all assumptions of Theorem 3 are verified. The remaining part of this proof
is thus devoted to prove 11. Let us consider x
1
, x
2
∈ B[0, R] and controls 0, w
n
∈ Ŵ T
such that, setting t
n
, x
n
. = t, y
n T ,x
1
[0, w
n
]·, we have V
e
n
T, x
1
≥ Z
1
l
n
t
n
, x
n
, 0, w
n
s ds + g
n
x
n
1 − ǫ . Hence, setting ˜t
n
, ˜ x
n
. = t, y
T ,x
2
[0, w
n
]· and noticing that ˜t
n
s = t
n
s = T ∀s ∈ [0, 1], we have
V
e
T, x
2
− V
e
n
T, x
1
≤ Z
1
lT, ˜x
n
, 0, w
n
s ds + g
n
˜ x
n
1 −
Z
1
l
n
T, x
n
, 0, w
n
s ds − g
n
x
n
1 + ǫ ≤
Z
1
|w
n
s|
β
[ǫn + ρ
l
| ˜ x
n
s − x
n
s|] ds +ρ
g
| ˜ x
n
1 − x
n
1| + ǫn + ǫ , where ǫn, ρ
l
and ρ
g
see A
3
and A
5
are determined with reference to the compact subset Q = B[0, R
′
]. If L
R
′
is the determination of L in A
e
1
for B[0, R
′
] then | ˜x
n
s − x
n
s| ≤ |x
1
− x
2
| + ǫnT + 1K
R
+ 1e
L
R′
T +1K
R
+ 1
. This, together with the fact that a similar inequality can be proved in a similar way when the
roles of V
e
and V
e
n
are interchanged, implies |V
e
T, x
2
− V
e
n
T, x
1
| ≤ K
R
ρ
l
|x
1
− x
2
| + ǫnT + 1K
R
+ 1e
L
R′
T +1K
R
+ 1
i + ρ
g
h |
x
1
− x
2
| + ǫnT + 1K
R
+ 1e
L
R′
T +1K
R
+ 1
i + K
R
+ 1ǫn . 12
144 F. Rampazzo – C. Sartori
Now, for τ ≤ T , let us estimate the difference V
e
n
τ, x − V
e
T, x, assuming that this difference is non negative. Let us set t
n
, x
n
· .
= t, y
n τ,
x
˜ w
, 0· with ˜
w s
. = T −
τ
1 β
∀s ∈ [0, 1]. Then the Dynamic Programming Principle V
e
n
τ, x − V
e
T, x ≤ Z
1
l
n
t
n
, x
n
, ˜ w
, 0s ds + V
e
n
T, x
n
1 − V
e
T, x . If M
. = max{M
1
+ M
2
, 1}, by A
e
2
we have |x
n
1 − x| ≤ M1 + R
′
| T − τ |. Hence, if
K
′ R
≥ max
t,x ∈[0,T ]×B[0,R
′
]
n∈
l
n
t, x, 1, 0, by the positive homogeneity of l
n
and by the first part of the proof we obtain
V
e
n
τ, x − V
e
T, x ≤ K
′ R
|T − τ | + σ
n
| T − τ |
13 where
σ
n
s =K
R
ρ
l
[M1 + R
′
s + ǫnT + 1K
R
+ 1e
L
R′
T +1K
R
+ 1
] + ρ
g
[M1 + R
′
s + ǫnT + 1K
R
+ 1e
L
R′
T +1K
R
+ 1
] + K
R
+ 1ǫn . Now let us estimate the difference V
e
T, x−V
e
n
τ, x, assuming it non negative. Let us consider
a sequence of controls w
n
, w
n
∈ Ŵτ such that, setting t
n
, x
n
. = t, y
n τ,
x
[w
n
, w
n
]·, one has
V
e
n
τ, x ≥
Z
1
l
n
t
n
, x
n
, w
n
, w
n
s ds + g
n
x
n
1 − ǫ . Then the controls 0, w
n
belong to ŴT , and, setting ˜t
n
, ˜ x
n
. = t, y[0, w
n
]·, we obtain V
e
T, x − V
e
n
τ, x ≤
Z
1
l˜t
n
, ˜ x
n
, 0, w
n
s ds + g ˜ x
n
1 −
Z
1
l
n
t
n
, x
n
, w
n
, w
n
s ds − g
n
x
n
1 + ǫ 14
for every n ∈ . Now one has
|x
n
s − ˜ x
n
s| ≤ Z
1
| f
n
t
n
, x
n
, w
n
, w
n
s − f ˜t
n
, ˜ x
n
, 0, w
n
s| ds ≤
Z
1
| f
n
t
n
, x
n
, w
n
, w
n
s − f t
n
, x
n
, w
n
, w
n
s| ds +
Z
1
| f t
n
, x
n
, w
n
, w
n
s − f ˜t
n
, x
n
, w
n
, w
n
s| ds +
Z
1
| f ˜t
n
, x
n
, w
n
, w
n
s − f ˜t
n
, ˜ x
n
, w
n
, w
n
s| ds +
Z
1
| f ˜t
n
, ˜ x
n
, w
n
, w
n
s − f ˜t
n
, ˜ x
n
, 0, w
n
s| ds . 15
for all s ∈ [0, 1]. In view of the parameter-free character of the system see e.g. [7] for the case α = β =
1, it is easy to show that one can transform the integral bound 10 into the pointwise bound
|w , w
s| ≤ ˜ K
R
∀s ∈ [0, 1] ,
On perturbation 145
where ˜ K
R
is a constant depending on R. Therefore, in view of basic continuity properties of the composition operator, there exists a modulus ρ such that the last integral in the above inequality
is smaller than or equal to ρ|T − τ |. Therefore, applying Gronwall’s inequality to 15 we obtain
|x
n
s − ˜ x
n
s| ≤T + 1K
R
+ 1[ǫn + ρ
f
| T − τ |
+ ρ|T − τ |]e
L
R′
T +1K
R
+ 1
. 16
Hence 14 yields V
e
T, x − V
e
n
τ, x ≤
Z
1
|l˜t
n
, ˜ x
n
, 0, w
n
s − l˜t
n
, x
n
, 0, w
n
s| ds +
Z
1
|l˜t
n
, x
n
, 0, w
n
s| ds − lt
n
, x
n
, 0, w
n
s| ds +
Z
1
|lt
n
, x
n
, 0, w
n
s − lt
n
, x
n
, w
n
, w
n
s ds +
Z
1
|lt
n
, x
n
, w
n
, w
n
s − l
n
t
n
, x
n
, w
n
, w
n
s| ds + ρ
g
| x
n
1 − ˜ x
n
1| . 17
Again, an argument based on the continuity properties of the composition operator allows one to conclude that there exists a modulus ˜
ρ such that
Z
1
|lt
n
, x
n
, 0, w
n
s − lt
n
, x
n
, w
n
, w
n
s| ds ≤ ˜ ρ|
T − τ | Therefore, plugging 15 into 16, we obtain
V
e
T, x − V
e
n
τ, x ≤ P
R
ρ
l
+ ρ
g
[ P
R
ǫ n + ρ
f
| T − τ | + ρ|T − τ |e
L
R′
P
R
] + P
R
[ρ
l
| T − τ | + ǫn] + ˜
ρ| T − τ | ,
18 where P
R
. = T + 1K
R
+ 1. Estimates 12, 13 and 18 imply the claim, so the theorem is proved.
5. Implementing optimal controls in the presence of perturbations
