August 2012
Intermediate to advanced
392 pages
10h 50m
English
Content preview from Nonlinear Optimal Control Theory
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274 Nonlinear Optimal Control Theory
Using the method of characteristics
p(r, t) =
p
0
(r − t) exp
−
R
r
r−t
µ(ρ)dρ
×exp
−
R
t
0
[f(N(s)) + u(s)]ds
, r > t
p(0, t − r) exp
−
R
r
0
µ(ρ)dρ
×exp
−
R
t
t−r
[f(N(s)) + u(s)]ds
, r < t
(8.8.3)
We have
p(0, t) = β
Z
t
0
k(r)p(0, t −r) exp
−
Z
r
0
µ(ρ)dρ
(8.8.4)
× exp
−
Z
t
t−r
[f(N(s)) + u(s)]ds
dr
+ β
Z
∞
t
k(r)p
0
(r −t, 0) exp
−
Z
r
r−t
µ(ρ)dρ
× exp
−
Z
t
0
[f(N(s)) + u(s)]ds
dr
Let
w(t) = p(0, t) exp
Z
t
0
[f(N(s)) + u(s)]ds
(8.8.5)
Then,
w(t) = β
Z
t
0
k(r)w(t − r) exp
−
Z
r
0
µ(ρ)dρ
dr (8.8.6)
+ β
Z
∞
t
k(r)p
0
(r − t) exp
−
Z
r
r−t
µ(ρ)dρ
dr,
N(t) =
Z
∞
0
p(r, t)dr (8.8.7)
= exp
−
Z
t
0
[f(N(s)) + u(s)]ds
"
Z
t
0
w(t −r) exp
−
Z
r
0
µ(ρ)dρ
dr
+
Z
∞
t
p
0
(r − t, 0) exp
−
Z
r
r−t
µ(ρ)dρ
dr
#
Let
η(t) =
Z
t
0
w(t − r) exp
−
Z
t
0
µ(ρ)dρ
dr (8.8.8)
+
Z
∞
t
p
0
(r − t) exp
−
Z
r
r−t
µ(ρ)dρ
dr
Examples 275
Then ...
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I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.