Nonlinear Optimal Control Theory

30 Nonlinear Optimal Control Theory

If we set

F

i

(t, x, y, x

′

, y

′

) = x

′

i

− f

i

(t, x, y

′

) i = 1, . . . , n, (2.6.4)

then the c ontrol problem can be written as the following proble m of Bo lza

in (n + m + 1 )- dimens ional (t, x, y)-space. The class of admissible arcs is the

set of absolutely c ontinuous functions

b

φ = (φ, η) = (φ

1

, . . . , φ

n

, η

1

, . . . , η

m

)

defined on intervals [t

0

, t

1

] such that:

(i) (t, φ(t), η

′

(t)) is in the do main of definition of the function

e

f = (f

0

, f);

(ii) (t

0

, φ(t

0

), t

1

, φ(t

1

)) is in B and η(t

0

) = 0;

(iii) the function t → f

0

(t, φ(t), η

′

(t)) is integrable; and

(iv)

F (t,

b

φ(t),

b

φ

′

(t)) = φ

′

(t) − f(t, φ(t), η

′

(t)) = 0 (2.6.5)

a.e. on [t

0

, t

1

]. The problem is to minimize the functional

g(t

0

, φ(t

0

), t

1

, φ(t

1

)) +

Z

t

1

t

0

f

0

(t, φ(t), η

′

(