Nonlinear Optimal Control Theory

174 Nonlinear Optimal Control Theory

−(1 − z)x and f(t, x, z) = zx, s o that f

0

and f satisfy the hypotheses of

Theorem 4.4.2. It follows from the state equations and the initial condition

(6.4.2) that for any control u s atisfying (6.4.3), the corr esponding trajectory

will sa tisfy

c ≤ φ(t) ≤ ce

t

. (6.4.4)

Hence the cons traint x ≥ 0 is always satisfied and so can be omitted from

further consideration. Since t

1

= T , (6.4.4) shows that all trajectories lie in a

compact set. The sets Q

+

(t, x) in this problem are given by

Q

+

(t, x) = {(y

0

, y): y

0

≥ (z − 1)x, y = zx, 0 ≤ z ≤ 1}

= {(y

0

, y): y

0

≥ y − x, 0 ≤ y ≤ x}.

For each (t, x) the set Q

+

(t, x) is closed and ...