Optimal Estimation of Dynamic Systems, 2nd Edition

Optimal Control and Estimation Theory 523

to the desired final conditions

θ

(t

f

)=

θ

f

˙

θ

(t

f

)=

˙

θ

f

The loss function to be minimized is given by

J =

1

2



t

f

t

0

u

2

(t) dt

where this J is not to be confused with the inertia. We restrict attention to the case

that t

0

= 0andt

f

= T are fixed. Two methods are considered to derive the optimal

maneuver. First we note that direct substitution of the dynamics equation into the

loss function yields an equation of the form given by

ϑ

(

θ

,

˙

θ

,

¨

θ

, t)=

1

2

¨

θ

2

(t)

This form is not identical to the form presented in Equation (8.1); however, the exten-

sion of the Euler-Lagrange equations to higher-order derivatives is straightforward

(which ...