Nonlinear Optimal Control Theory

Examples 279

FIGURE 8.2

FIGURE 8.3

280 Nonlinear Optimal Control Theory

FIGURE 8.4

If |ψ

2

(t

0

)| ≤ λ

3

, the possible sequence of values assumed by u are

(a) {0} (c) {0, −1}

(b) {0, 1}

If λ

3

= 0, we immediately see from (8.9.8) that u takes values 1 or −1, except

at a single point which we can neglect. That is, the sequence of control values

assumed ar e

(a) {−1, 1}

(b) {1, −1}

When λ

3

= 0 the trajectory pattern and control synthesis could be deduced

from Fig. 8.4. Note that as λ

2

→ 0 the lines x

2

=

p

λ

1

/λ

2

and x

2

= −

p

λ

1

/λ

2

pass to infinity and the trajectory pattern g e ...