Nonlinear Optimal Control Theory

20 Nonlinear Optimal Control Theory

(ii) φ is a solution of the system o f differential equations

dx

dt

= f(t, x, u(t)); (2.3.1)

that is,

φ

′

(t) = f(t, φ(t), u(t)) a.e. on [t

0

, t

1

].

The point (t

0

, φ(t

0

)) w ill be called the initial point of the trajectory

and the point (t

1

, φ(t

1

)) will be called the terminal point of the trajec-

tory. The point (t

0

, φ(t

0

), t

1

, φ(t

1

)) will be called the en d point of the

trajectory.

Note that since φ is a bsolutely continuous, it is the integral of its derivative.

Hence (ii) contains the statement that the function t → f(t, φ(t), u(t)) is

Leb esgue integrable on [t

0

, t

1

].

The system of differential equations (2.3.1) will be