Nonlinear Optimal Control Theory

330 Nonlinear Optimal Control Theory

We may write

ν

0t

=

n+1

X

i=1

Π

0

i

(t)δ

u

0

i

(t)

, Π

0

i

≥ 0,

X

Π

0

i

= 1,

and thus,

F

0

(φ

0

(t), ν

0t

, t) =

X

Π

0

i

(t)f

0

(φ

0

(t), u

0

i

(t), t).

Similarly,

F (φ

0

(t), ν

0t

, t) =

X

Π

0

i

(t)f(φ

0

(t), u

0

i

(t), t).

Then, we write

u

0

(t) ≡ (Π

0

1

(t), . . . , Π

0

n+1

(t), u

0

1

(t), . . . , u

0

n+1

(t)),

Π

0

i

≥ 0,

X

Π

0

i

= 1, u

0

i

∈ Ω,

Remark 11.8.3. In general, in this section, when we write u(t), we think of

it in the form

u(t) ≡ (Π

1

(t), . . . , Π

n+1

(t), u

1

(t), . . . , u

n+1

(t)),

Π

i

≥ 0,

X

Π

i

= 1, u

i

∈ Ω.

Then,

F

0

(φ(t), u(t), t) ≡

X

Π

i

(t)f

0

(φ(t), u

i

(t), t),

F (φ(t), u(t), t) ≡

X

Π

i

(t)f(φ(t), u

i

(t), t).

Now, in (i) above we assume λ

0

= 1, and let

e

H(φ, u, t) = F

0

(φ(t), u(t), t) − (Φ(t) − λ(t)∇G(φ(t))) · F (φ(t), u(t), t)

−

˙

Φ · φ + λ(0

+

)G(φ(0))/t

1

− (Φ(0) ·φ(0) −Φ(t

1

) · φ(t

1

))/t

1

+ [β

1

T (φ(0)) + β