Chemistry Reference
In-Depth Information
1
p
=−
(
∂H/∂x
)
=
2
(
∂L/∂x
), one deduces that
d
dt
∂L
∂ x
∂L
∂x
=
which is the Euler-Lagrange equation.
3.
The Time-Minimal Control Problem
Now, we particularize the PMP to the time-minimal control problem.
Proposition 1
Consider the time-minimal control problem for a system of the
form:
x
m
.If
(
x, u
)
is an
optimal solution on
[0
,T
]
then there exists a nonzero adjoint vector
p
(
t
)
such that
the following equations are satisfied:
=
F
(
x, u
)
, where the control domain is a subset
U
of
R
dx
dt
=
∂H
∂p
,
dp
dt
=−
∂H
∂x
H
(
x, p, u
)
=
M
(
x, p
)
where
H
=
p
·
F
(
x, u
)
is the pseudo-Hamiltonian of the system and
M
(
x, p
)
=
max
v
∈
U
H
(
x, p, v
)
. Moreover,
M
is constant and non-negative.
Note that in this statement of the PMP, the constant
p
0
has been substracted in the
definition of the pseudo-Hamiltonian using the fact that this Hamiltonian is equal
to 0 when there is no constraint on the control duration.
Proposition 2
Consider an affine control system of the form
m
x
=
F
0
(
x
)
+
u
i
F
i
(
x
)
,
|
u
|≤
1
i
=
1
Then, outside the surface
:
H
i
=
0
,i
=
1
,...,m,
the optimal solutions are
(
i
=
1
,m
H
i
)
1
/
2
, where
H
i
=
given by the Hamiltonian
H
r
=
H
0
+
p
·
F
i
(
x
)
is
the Hamiltonian corresponding to the vector field
F
i
.
Proof
We apply the maximum principle for the time-minimal control prob-
lem with control bound
|
u
|≤
1. The pseudo-Hamiltonian takes the form
H
=
H
0
+
i
u
i
H
i
and the control domain
is the unit b
all
|
u
|≤
1. The maximization
H
i
/
i
=
1
,m
H
i
. Plugging such control into
H
defines the normal hamiltonian
H
r
.
condition gives outside
:
u
i
=
