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
1
-A
5
.
i For every t, x
∈ [0, T [×
k
and for every n ∈
one has V
e
t, x = V t, x; and
V
e
n
t, x = V
n
t, x;
ii the maps V
e
and V
e
n
are continuous on [0, T ] ×
k
. Thanks to this theorem – which, in particular, implies that V and V
n
can be continuously extended on [0, T ]
×
k
– the problem of the convergence of the V
n
is transformed in the analogous problem for the V
e
n
. We now recall that each of these value functions is the unique solution of a suitable boundary
value problem. This is a consequence of the comparison theorem below. To state these results, let us introduce the extended Hamiltonians
H
e
t, x, p ,
p .
= sup
w ,w∈
[0,+∞[×
m
∩ S
+ m
{−p w
β
− hp, f t, x, w , w
i − lt, x, w , w
} 6
where S
+ m
. = {w
, w ∈ [0, +∞[×
m
: |w
, w | = 1},
H
e
n
t, x, p ,
p .
= sup
w ,w∈
[0,+∞[×
m
∩ S
+ m
{−p w
β
−hp, f
n
t, x, w , w
i−l
n
t, x, w , w
} , and the corresponding Hamilton-Jacobi-Bellman equations
H J
e
H
e
t, x, u
t
, u
x
= 0 , H J
e
n
H
e
n
t, x, u
t
, u
x
= 0 . For the sake of self consistency let us recall the definition of possibly discontinuous vis-
cosity solution, which was introduced by H. Ishii in [5]. Given a function F :
✁
→ ,
✁
⊆
k
, let us consider the upper and lower semicontinuous envelopes, defined by
F
∗
x .
= lim
r→0
+
sup {Fy : y ∈
✁
, |x − y| ≤ r} ,
F
∗
x .
= lim
r→0
+
inf {Fy : y ∈
✁
, |x − y| ≤ r} ,
x ∈
✁
, respectively. Of course, F
∗
is upper semicontinuous and F
∗
is lower semicontinuous. D
EFINITION
1. Let E be a subset of
s
and let G be a real map, the Hamiltonian, defined on E
× ×
s
. An upper[resp. lower]-semicontinuous function u is a viscosity subsolution [resp. supersolution] of
Gy, u, u
y
= 0 7
On perturbation 141
at y ∈ E if for every φ ∈
1 s
such that y is a local maximum [resp. minimum] point of u − φ
on E one has G
∗
y, φ y, φ
y
y ≤ 0
[resp. G
∗
y, φ y, φ
y
y ≥ 0] .
A function u is a viscosity solution of 7 at y ∈ E if u
∗
is a viscosity subsolution at y and u
∗
is a viscosity supersolution at y.
T
HEOREM
3 C
OMPARISON
. Assume A
1
-A
5
. Let u
1
: [0, T ] ×
k
→ be an upper
semicontinuous, bounded below, viscosity subsolution of H J
e
in ]0, T [ ×
k
, continuous on {0} ×
k
∪ {T } ×
k
. Let u
2
: [0, T ] ×
k
→ be a lower semicontinuous, bounded below,
viscosity supersolution of H J
e
in [0, T [ ×
k
. For every x ∈
k
, assume that
u
1
T, x ≤ u
2
T, x or
u
2
is a viscosity supersolution of H J
e
at T, x . Then
u
1
t, x ≤ u
2
t, x ∀t, x ∈ [0, T ] ×
k
. The same statement holds true for the equations H J
e
n
. As a consequence of this theorem and of a suitable dynamic programming principle for the
extended problems one can prove the following: T
HEOREM
4. The value function V
e
is the unique map which i is continuous on
{0} ×
k
∪ {T } ×
k
; ii is a viscosity solution of H J
e
in ]0, T [ ×
k
; iii satisfies the following mixed type boundary condition:
BC
e
m
V
e
T, x ≤ gx ∀x ∈
k
and
V
e
T, x = gx
or V
e
is a viscosity supersolution of H J
e
at T, x . Once we replace H J
e
by H J
e
n
, the same statement holds true for the maps V
e
n
. Finally let us recall a regularity result which will be useful in the proof of Theorem 1.
T
HEOREM
5. Assume A
1
-A
5
and fix R 0. Then there exists R
′
≥ R and positive con- stants C
1
, C
2
such that |V
e
t, x
1
− V
e
t, x
2
| ≤ C
1
ρ
l
C
2
|x
2
− x
1
| + ρ
g
C
2
|x
2
− x
1
| for every t, x
1
t, x
2
∈ [0, T ] × B[0; R], where ρ
l
and ρ
g
are the modulus appearing in A
3
and the modulus of uniform continuity of g, respectively, corresponding to the compact [0, T ] ×
B[0 ; R
′
]. Moreover for every t ∈ [0, T [ one has
|V
e
t, x − V
e
t , x | ≤ η
t
|t − t|
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
