Nonlinear Optimal Control Theory

Examples of Control Problems 11

Thus, the time of traverse T along C is given by

T =

1

(2g)

1/2

Z

x

1

x

0



1 + (y

′

)

2

y − α



1/2

dx.

The problem of finding a curve C that minimizes the time of traverse is

that of finding a function y = y(x) that minimizes the integ ral

Z

x

1

x

0



1 + (y

′

)

2

y − α



1/2

dx. (1.6.3)

Note that if v

0

= 0, then the integral is improper.

We put this problem in a format similar to the previous ones as follows.

Change the notation for the independent variable fr om x to t. Then set

y

′

= u y(t

0

) = y

0

. (1.6.4)

A continuous function u will be called admissible if it is defined on [t

0

, t

1

],

if the solution of (1.6.4) corresponding to u satisfies y(t

1

) = y

1

,