Nonlinear Optimal Control Theory

Formulation of Control Problems 25

The following observation will be useful in the sequel. Since for a ny admis-

sible solution of the fixed time problem we have ω(s) = t

1

− t

0

> 0, we can

take the set

¯

R for the fixe d end-time problem to be [0, 1] × R × R

+

, where

R

+

= {w : w > 0}.

A s pecial case of the end conditions occurs if the initial and terminal data

are sepa rated. In this event, a set T

0

of points (t

0

, x

0

) in R

n+1

and a set T

1

of

points (t

1

, x

1

) in R

n+1

are g iven and an admissible trajectory is required to

satisfy the conditions

(t

i

, φ(t

i

)) ∈ T

i

, i = 0, 1. (2.4.8)

The set B in this case is given by (2.2.4). We shall show that the apparently

more general ...