| dc.description.abstract | In this thesis, we study value functions of a class of optimal control and differential game problems which satisfy certain quasivariational inequalities (QVI). We study the smoothness properties of the value functions and characterize them as the unique viscosity solutions of the associated QVIs.
Such a prototype control problem is impulse control. It is known that the value function of an impulse control problem with bounded costs and dynamics is the unique viscosity solution of the associated QVI in the class of bounded uniformly continuous functions,
??
??
??
(
??
??
)
BUC(R
d
). We relax the boundedness assumptions on the cost function and prove uniqueness for the infinite horizon problem using the uniqueness of stopping time problems in the uniformly continuous function class,
??
??
(
??
??
)
UC(R
d
).
The trajectory of the impulse control problem in infinite horizon is given by
??
(
??
)
=
??
(
??
??
)
+
??
(
??
(
??
??
)
,
??
??
)
,
??
?
[
??
??
,
??
??
+
1
)
,
x(t)=x(?
i
?
)+f(x(?
i
?
),u
i
?
),t?[?
i
?
,?
i+1
?
),
where
??
(
0
)
=
??
0
x(0)=x
0
?
,
{
??
??
}
??
?
??
{?
i
?
}
i?N
?
is such that
??
??
?
?
?
n
?
?? as
??
?
?
n??,
??
:
??
??
×
??
?
??
??
f:R
d
×U?R
d
with
??
U a compact metric space, and the impulse
??
=
{
??
??
}
??
?
??
?={?
i
?
}
i?N
?
is a sequence in
(
??
+
)
??
(R
+
)
d
. The control
??
(
??
)
:
[
0
,
?
)
?
??
u(t):[0,?)?U is any measurable function, and
??
=
(
??
,
??
,
??
(
?
)
)
?=(?,?,u(?)) is the control variable.
The optimal cost functional is defined by
??
(
??
)
:
=
inf
?
??
{
?
0
?
??
(
??
(
??
)
,
??
(
??
)
)
??
?
??
??
??
??
+
?
??
=
0
?
??
(
??
(
??
??
)
,
??
??
)
??
?
??
??
??
}
,
V(x):=
?
inf
?
{?
0
?
?
k(x(t),u(t))e
??t
dt+
i=0
?
?
?
c(x(?
i
?
),?
i
?
)e
???
i
?
},
where
??
k is the running cost,
??
(
??
,
??
)
c(x,?) is the impulse cost, and
??
? is the discount factor. Using the dynamic programming principle (DPP), we can show that
??
V satisfies the following QVI in the viscosity sense:
max
?
{
??
+
??
(
??
,
??
??
)
,
??
?
??
??
}
=
0
,
?
??
?
??
??
,
max{V+H(x,DV),V?MV}=0,?x?R
d
,
where
??
??
(
??
)
=
inf
?
??
{
??
(
??
+
??
)
+
??
(
??
,
??
)
}
,
??
?
??
??
(
??
??
)
,
MV(x)=
?
inf
?
{V(x+?)+c(x,?)},??UC(R
d
),
and
??
(
??
,
??
)
=
sup
?
??
?
??
{
?
??
(
??
,
??
)
?
??
?
??
(
??
,
??
)
}
.
H(x,p)=
u?U
sup
?
{?f(x,u)?p?k(x,u)}.
Theorem 1: Assume
??
f is bounded and globally Lipschitz continuous in the first variable with Lipschitz constant
??
L,
??
>
??
?>L, and
??
,
??
k,c are uniformly continuous with
??
1
(
1
+
?
??
?
)
?
??
(
??
,
??
)
?
??
2
(
1
+
?
??
?
)
,
?
??
,
??
?
??
??
,
C
1
?
(1+???)?c(x,?)?C
2
?
(1+???),?x,??R
d
,
and
??
(
??
,
??
+
??
)
<
??
(
??
,
??
)
+
??
(
??
,
??
)
.
c(x,?+?)<c(x,?)+c(x,?).
Then
??
V defined above is the unique viscosity solution of the QVI in
??
??
(
??
??
)
UC(R
d
).
We also study the impulse control problem over finite horizon
??
>
0
T>0. For fixed
??
T, the trajectory evolves according to the same dynamics until time
??
T. The optimal cost functional
??
(
??
,
??
)
V(s,x) is
??
(
??
,
??
)
=
inf
?
??
{
?
??
??
??
(
??
(
??
)
,
??
(
??
)
)
??
??
+
?
??
?
??
??
?
??
??
(
??
(
??
??
)
,
??
??
)
+
??
(
??
(
??
)
)
}
,
V(s,x)=
?
inf
?
{?
s
T
?
k(x(t),u(t))dt+
s??
i
?
?T
?
?
c(x(?
i
?
),?
i
?
)+g(x(T))},
where
??
g is the terminal cost.
Then
??
V satisfies the QVI in the viscosity sense:
max
?
{
??
??
+
??
(
??
,
??
??
)
,
??
?
??
??
}
=
0
in
(
0
,
??
)
×
??
??
,
max{V
t
?
+H(x,DV),V?MV}=0in (0,T)×R
d
,
??
(
??
,
??
)
=
??
(
??
)
,
V(T,x)=g(x),
with
??
??
(
??
,
??
)
=
inf
?
??
{
??
(
??
,
??
+
??
)
+
??
(
??
,
??
)
}
,
??
(
??
,
??
)
=
sup
?
??
?
??
{
?
??
(
??
,
??
)
?
??
?
??
(
??
,
??
,
??
)
}
.
MV(t,x)=
?
inf
?
{V(t,x+?)+c(t,?)},H(x,p)=
u?U
sup
?
{?f(x,u)?p?k(t,x,u)}.
Theorem 2: Assume
??
f is bounded and globally Lipschitz in the first variable,
??
k and
??
g are uniformly continuous, and
??
c is continuous and bounded away from zero. Then the QVI has a unique solution in
??
??
(
[
0
,
??
]
×
??
??
)
UC([0,T]×R
d
).
If
??
f is only locally Lipschitz:
?
??
(
??
,
??
)
?
??
(
??
,
??
)
?
?
??
??
?
??
?
??
?
,
?
??
?
,
?
??
?
<
??
,
?
??
(
??
,
??
)
?
?
??
(
1
+
?
??
?
)
,
?f(x,u)?f(y,u)??L
R
?
?x?y?,?x?,?y?<R,?f(x,u)??F(1+?x?),
the value function lies in
??
(
??
??
×
[
0
,
??
]
)
C(R
d
×[0,T]) and is not necessarily uniformly continuous.
Theorem 3: Under these conditions, the QVI has a unique solution in
??
(
??
??
×
[
0
,
??
]
)
C(R
d
×[0,T]).
In Chapter 3, we study a hybrid optimal control problem, where the trajectory is affected by both continuous and discrete controls. The trajectory evolves in different Euclidean spaces depending on the hitting of autonomous jump sets
??
A or controlled sets
??
C, with discrete jumps governed by maps
??
g. The total discounted cost is
??
(
??
)
=
inf
?
??
?
??
=
0
?
?
??
??
??
??
+
1
??
(
??
(
??
)
,
??
(
??
)
)
??
??
+
??
??
(
??
(
??
??
)
,
??
??
)
+
??
??
(
??
(
??
??
)
,
??
??
)
,
V(x)=
?
inf
?
i=0
?
?
?
?
?
i
?
?
i+1
?
?
k(x(t),u(t))dt+c
a
?
(x(?
i
?
),v
i
?
)+c
c
?
(x(?
i
?
),?
i
?
),
where
??
??
c
a
?
is the autonomous jump cost and
??
??
c
c
?
is the controlled jump cost.
Theorem 4: Under suitable transversality and compactness conditions, the value function
??
V is locally Hölder continuous and satisfies the DPP and an associated QVI in the viscosity sense.
Theorem 5: The QVI has a unique viscosity solution.
In Chapter 4, we study differential games with hybrid controls: the minimizer uses continuous, switching, and impulse controls, while the maximizer uses continuous and switching controls. Let
??
??
U
i
?
be compact control sets,
??
??
D
i
?
switching sets, and
??
K the impulse control set. The dynamics are
??
(
??
)
=
??
(
??
(
??
)
,
??
1
(
??
)
,
??
1
(
??
)
,
??
2
(
??
)
,
??
2
(
??
)
)
+
?
impulses
,
??
(
0
?
)
=
??
,
X(t)=f(X(t),u
1
?
(t),d
1
?
(t),u
2
?
(t),d
2
?
(t))+?impulses,X(0?)=x,
with running costs
??
k, switching costs
??
??
c
i
?
, and impulse costs
??
I. Using Elliott-Kalton non-anticipating strategies, we define the upper and lower value functions:
??
?
(
??
)
=
inf
?
??
sup
?
??
??
(
??
,
??
[
??
]
,
??
)
,
??
+
(
??
)
=
sup
?
??
inf
?
??
??
(
??
,
??
,
??
[
??
]
)
.
V
?
(x)=
?
inf
?
?
sup
?
J(x,?[?],?),V
+
(x)=
?
sup
?
?
inf
?
J(x,?,?[?]).
Theorem 6: If the Isaacs min-max condition holds (
??
?
=
??
+
H
?
=H
+
), then
??
?
=
??
+
V
?
=V
+
and is the unique viscosity solution of the associated Hamilton-Jacobi-Isaacs (HJI) system.
In Chapter 5, we extend the hybrid control model to zero-sum differential games, allowing the minimizing player continuous, autonomous, and controlled jumps, while the maximizer has continuous and autonomous jumps. The discounted cost functional is
??
(
??
,
??
1
,
??
2
,
??
1
,
??
2
)
=
?
0
?
??
(
??
(
??
)
,
??
1
,
??
2
)
??
?
??
??
??
??
+
?
??
??
??
(
.
.
.
)
??
?
??
??
??
+
?
??
??
??
(
.
.
.
)
??
?
??
??
??
.
J(x,u
1
?
,u
2
?
,?
1
?
,v
2
?
)=?
0
?
?
k(X(t),u
1
?
,u
2
?
)e
??t
dt+
i
?
?
c
a
?
(...)e
???
i
?
+
i
?
?
c
c
?
(...)e
???
i
?
.
Defining Elliott-Kalton strategies, the upper and lower value functions are Hölder continuous, satisfy DPP, and their respective QVIs have unique viscosity solutions. Under Isaacs-like min-max conditions, the differential game has a well-defined value | |
