12.3 Optimal Control Theory 681
Differentiation of Eq. (E 5 ) leads to
2 u ̇+ ̇λ= 0 (E 6 )
Equations(E 4 ) and (E 6 ) yield
u ̇=x (E 7 )
Sincex ̇=u[Eq. (E 2 )], we obtain
x ̈=u ̇=x
that is,
x ̈−x= 0 (E 8 )
The solution of Eq. (E 8 ) is given by
x(t)=c 1 sinht+c 2 cosht (E 9 )
wherec 1 andc 2 are constants. By using the initial conditionx(0)=1, we obtainc 2 =. 1
Sincexis not fixed at the terminal pointt=T=1, we use the conditionλ=0 at
t=1 in Eq. (E 5 ) and obtainu(t=1)=0. But
u= ̇x=c 1 cosh t+sinht (E 10 )
Thus
u( 1 )= 0 =c 1 cosh 1 +sinh 1
or
c 1 =
−sinh 1
cosh 1
(E 11 )
and hence the optimal control is
u(t)=
−sinh 1
cosh 1
·cosht+sinht
=
−sinh 1·cosht+cosh 1·sinht
cosh 1
=
−sinh( 1 −t)
cosh 1
(E 12 )
The corresponding state trajectory is given by
x(t)= ̇u=
cosh( 1 −t)
cosh 1
(E 13 )
12.3.2 Necessary Conditions for a General Problem
We shall now consider the basic optimal control problem stated earlier:
Find the optimal control vectoruthat minimizes
J=
∫T
0
f 0 ( x,u,t) dt (12.35)
subject to
x ̇i=fi( x,u,t), i= 1 , 2 ,... , n (12.36)
