August 2012
Intermediate to advanced
392 pages
10h 50m
English
Content preview from Nonlinear Optimal Control Theory
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84 Nonlinear Optimal Control Theory
Henceforth we shall simplify our notation by writing the preceding integral as
follows:
f(t, x, µ
t
) ≡
Z
Ω(t)
f(t, x, z)dµ
t
. (4.3.1)
Thus, a relaxed trajectory corres ponding to a control µ will be a solution of
the differential equation
x
′
= f(t, x, µ
t
). (4.3.2 )
We shall use Greek letters to denote relaxed controls. The subscript t denotes
the probability measure µ
t
on Ω(t), that is, the value of µ at t. The subscript
notation is used to e mphasize that f(t, x, µ
t
) is defined by (4.3.1).
In this section we shall be concerned with functions f = (f
1
, . . . , f
n
),
where e ach of the f
i
is re al valued and defined on a set I ×X ×U, where I is
a real compact interval, X is an open inter val in R
n
, and U is an open interval
in R
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