Nonlinear Optimal Control Theory

288 Nonlinear Optimal Control Theory

subject to:

d

dt

φ

i

(t) = f

i

(t, φ(t), u(t)) +

Z

t

0

g

i

(t, s, φ(s), u(s))ds (9.5.2)

u(t) ∈ Ω (t, φ(t)) (9.5.3)

T (φ(0), φ(t

1

)) = 0. (9.5.4)

For assumptions on f

0

, f

1

, . . . , f

n

refer to Assumption 6.3.1. For assumptions

on g

i

, i = 1, . . . , n refer to Section 2.7. T is continuously differentiable.

Remark 9.5.1. Under Assumption 9.2.1 and additional assumptions on

the kernel functions, we can rewrite (9.2.1) to (9.2.4) to fit the form of

(9.5.1) to (9.5.4). We may introduce a new state variable ψ such that

ψ

′

(t) = M (φ(t), u(t), t), and consider the constraint [W (φ(t

1

)) + ψ(t

1

)]

2

+

[φ(0) − F (0)]

2

= 0. Finally we differentiate ...