In control theory, the minimum energy control is the control
u
(
t
)
that will bring a linear time invariant system to a desired state with a minimum expenditure of energy.
Let the linear time invariant (LTI) system be
x
˙
(
t
)
=
A
x
(
t
)
+
B
u
(
t
)
y
(
t
)
=
C
x
(
t
)
+
D
u
(
t
)
with initial state
x
(
t
0
)
=
x
0
. One seeks an input
u
(
t
)
so that the system will be in the state
x
1
at time
t
1
, and for any other input
u
¯
(
t
)
, which also drives the system from
x
0
to
x
1
at time
t
1
, the energy expenditure would be larger, i.e.,
∫
t
0
t
1
u
¯
∗
(
t
)
u
¯
(
t
)
d
t
≥
∫
t
0
t
1
u
∗
(
t
)
u
(
t
)
d
t
.
To choose this input, first compute the controllability gramian
W
c
(
t
)
=
∫
t
0
t
e
A
(
t
−
τ
)
B
B
∗
e
A
∗
(
t
−
τ
)
d
τ
.
Assuming
W
c
is nonsingular (if and only if the system is controllable), the minimum energy control is then
u
(
t
)
=
−
B
∗
e
A
∗
(
t
1
−
t
)
W
c
−
1
(
t
1
)
[
e
A
(
t
1
−
t
0
)
x
0
−
x
1
]
.
Substitution into the solution
x
(
t
)
=
e
A
(
t
−
t
0
)
x
0
+
∫
t
0
t
e
A
(
t
−
τ
)
B
u
(
τ
)
d
τ
verifies the achievement of state
x
1
at
t
1
.
