Engineering Optimization: Theory and Practice, Fourth Edition

682 Optimal Control and Optimality Criteria Methods

Now we introduce aLagrange multiplier pi, also known as theadjoint variable, for

theith constraint equation in (12.36) and form an augmented functionalJ∗as

J∗=

∫T

0

[

f 0 +

∑n

i= 1

pi(fi− ̇xi)

]

dt (12.37)

The Hamiltonian functional,H, is defined as

H=f 0 +

∑n

i= 1

pifi (12.38)

such that

J∗=

∫T

0

(

H−

∑n

i= 1

pix ̇i

)

dt (12.39)

Since the integrand

F=H−

∑n

i= 1

pix ̇i (12.40)

depends onx, u, andt, there aren+mdependent variables (xandu) and hence the

Euler–Lagrange equations become

∂F

∂xi

−

d

dt

(

∂F

∂x ̇i

)

= 0 , i= 1 , 2 ,... , n (12.41)

∂F

∂uj

−

d

dt

(

∂F

∂u ̇j

)

= 0 , j= 1 , 2 ,... , m (12.42)

In view of relation (12.40), Eqs. (12.41) and (12.42) can be rewritten as

−

∂H

∂xi

=pi, i= 1 , 2 ,... , n (12.43)

∂H

∂ui

= 0 , j= 1 , 2 ,... , m (12.44)

Equations (12.43) are knowns asadjoint equations.

The optimum solutions forx, u, andpcan be obtained by solving Eqs. (12.36),

(12.43), and (12.44). There are totally 2n+mequations withnxi’s, npi’s, and muj’s

as unknowns. If we know the initial conditionsxi(0),i= 1 , 2 ,... , n, and the terminal

conditionsxj(T ),j= 1 , 2 ,... , l, withl < n, we will have the terminal values of the

remaining variables, namelyxj(T ),j=l+ 1 ,l+ 2 ,... , n, free. Hence we will have

to use the free end conditions

pj(T ) = 0 , j=l+ 1 , l+ 2 ,... , n (12.45)

Equations (12.45) are called thetransversality conditions.