Nonlinear Optimal Control Theory

Bounded State Problems 335

Note that

˙

ψ

2

= −α sin θ

˙

ψ

2

˙

θ = −α(sin θ)

˙

θ

∴

˙

ψ

2

ψ

2

= −α(sin θ)

˙

θ

∴

d

dt



1

2

ψ

2

2

− α cos θ



= 0

∴

1

2

ψ

2

2

= α cos θ − α cos θ(0).

Thus, the cost becomes

J = 2α

Z

1

0

cos θ dt − α cos θ(0)

where

¨

θ + α sin θ = 0 (11.9.6)

˙

θ(0) =

˙

θ(1) = 0, |

˙

θ| ≤

α

20

Corresponding to the e quation,

¨

θ + α sin θ = 0.

We have the phase portra it shown in Fig. 11.1.

FIGURE 11.1

In (11.9.6) what we have is the familiar pendulum equation. For the undamped

pendulum any periodic orbit must intersect the θ-axis of the phase portrait

at two points, say (−b, 0) and (b, 0) and the period T is given by

T =

4

√

α

Z

π/2

0

dξ

q

1 − sin

2



b

2



sin

2

ξ

(11.9.7)

336 Nonlinear Optimal Control Theory

In