Biomedical Engineering Reference
In-Depth Information
m
(
·
)
R
+
→
R
controlled variables
u
:
that control the dynamical system. It can be
written formally as
T
f
0
g
0
min
u
(
·
)
,
T
(
t
,
x
(
t
)
,
u
(
t
))
d
t
+
(
T
,
x
(
T
))
0
x
(
t
)=
f
(
t
,
x
(
t
)
,
u
(
t
))
, ∀
t
∈
[
0
,
T
]
u
(
t
)
∈
U
t
, ∀
t
∈
[
0
,
T
]
x
(
0
)=
x
0
,
x
(
T
)
∈
M
1
(16)
Here,
f
0
is the cost function and
g
0
is the final cost. The final time
T
can be either
fixed or be part of the control. The dynamical system is represented by function
f
,
which gives the evolution of
x
and is controlled by
u
. At each time, the control may
be subject to constraints represented by the set
U
t
and the set
M
1
is a subset of
n
R
representing conditions on the final state. If we replace the constraints
x
(
0
)=
x
0
and
x
,wehavea
T
-periodic optimal control problem (see [
28
]
for more precision on the consequences of this model).
A major tool of optimal control is Pontryagin's maximum principle [
121
]. It
gives necessary optimality conditions for the optimal trajectories. We denote the
Hamiltonian of the system by
(
T
)
∈
M
1
by
x
(
T
)=
x
(
0
)
n
i
=
1
p
i
f
i
(
t
,
x
,
u
)
+
p
0
f
0
p
0
H
(
t
,
x
,
p
,
,
u
)=
(
t
,
x
,
u
)
,
p
1
p
n
n
and
p
0
where
p
=(
,...,
)
∈
R
∈
R
.If
u
(
·
)
associated with the trajectory
x
(
·
)
is an optimal control on
[
0
,
T
]
, then there exists a continuous application
p
(
·
)
called
the adjoint vector and a nonpositive number
p
0
such that for almost all
t
∈
[
0
,
T
]
,
)=
∂
H
)=
−
∂
H
∂
p
0
p
0
x
(
t
p
(
t
,
x
(
t
)
,
p
(
t
)
,
,
u
(
t
))
,
p
(
t
x
(
t
,
x
(
t
)
,
p
(
t
)
,
,
u
(
t
))
(17)
∂
and we have the maximisation condition for almost all
t
∈
[
0
,
T
]
,
p
0
p
0
H
(
t
,
x
(
t
)
,
p
(
t
)
,
,
u
(
t
)) =
max
v
H
(
t
,
x
(
t
)
,
p
(
t
)
,
,
v
)
.
(18)
∈
U
t
If in addition, the final time to reach the target
M
1
is not fixed, we have the condition
g
0
∂
p
0
∂
p
0
max
v
H
(
T
,
x
(
T
)
,
p
(
T
)
,
,
v
)=
−
t
(
T
,
x
(
T
))
∈
U
T
n
and if
M
1
is manifold of
R
with a tangent space T
x
M
1
at
x
,wehave
p
0
∂
g
p
(
T
)
−
t
(
T
,
x
(
T
))
⊥
T
x
(
T
)
M
1
.
∂
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