In optimal control, problems of singular control are problems that are difficult to solve because a straightforward application of Pontryagin's minimum principle fails to yield a complete solution. Only a few such problems have been solved, such as Merton's portfolio problem in financial economics or trajectory optimization in aeronautics. A more technical explanation follows.
The most common difficulty in applying Pontryagin's principle arises when the Hamiltonian depends linearly on the control
u
, i.e., is of the form:
H
(
u
)
=
ϕ
(
x
,
λ
,
t
)
u
+
⋯
and the control is restricted to being between an upper and a lower bound:
a
≤
u
(
t
)
≤
b
. To minimize
H
(
u
)
, we need to make
u
as big or as small as possible, depending on the sign of
ϕ
(
x
,
λ
,
t
)
, specifically:
u
(
t
)
=
{
b
,
ϕ
(
x
,
λ
,
t
)
<
0
?
,
ϕ
(
x
,
λ
,
t
)
=
0
a
,
ϕ
(
x
,
λ
,
t
)
>
0.
If
ϕ
is positive at some times, negative at others and is only zero instantaneously, then the solution is straightforward and is a bang-bang control that switches from
b
to
a
at times when
ϕ
switches from negative to positive.
The case when
ϕ
remains at zero for a finite length of time
t
1
≤
t
≤
t
2
is called the singular control case. Between
t
1
and
t
2
the maximization of the Hamiltonian with respect to u gives us no useful information and the solution in that time interval is going to have to be found from other considerations. (One approach would be to repeatedly differentiate
∂
H
/
∂
u
with respect to time until the control u again explicitly appears, which is guaranteed to happen eventually. One can then set that expression to zero and solve for u. This amounts to saying that between
t
1
and
t
2
the control
u
is determined by the requirement that the singularity condition continues to hold. The resulting so-called singular arc will be optimal if it satisfies the Kelley condition:
(
−
1
)
k
∂
∂
u
[
(
d
d
t
)
2
k
H
u
]
≥
0
,
k
=
0
,
1
,
⋯
. This condition is also called the generalized Legendre-Clebsch condition).
The term bang-singular control refers to a control that has a bang-bang portion as well as a singular portion.