Nonlinear Optimal Control Theory

Proof of t he Maximum Principle 215

Lemma 7. 4.5 (Euler Equation). Let Assumption 7.4.1 hold and let ψ

1

be a

function in the set T defined in (v) of Assumption 7.4.1. For fixed ε > 0 let

T

ε

(ψ

1

) = {ψ ∈ T : kψ

′

− ψ

′

1

k

2

< ε, |ψ(t

0

) − ψ

1

(t

0

)|

2

< ε}, (7.4.7)

where k k denotes the L

2

norm. Let ψ minimize (7.4.1) on the set T

ε

(ψ

1

). Let

G

x

(t) = G

x

(t, ψ(t), ψ

′

(t)) and let G

x

′

(t) have similar meaning. Then there

exists a constant c such that for a.e. t in [t

0

, t

1

]

G

x

′

(t) =

Z

t

t

0

G

x

(s)ds + c. (7.4.8)

Proof. From the Cauchy-Schwarz inequality and (7.4.7) we have that for all

ψ in T

ε

(ψ

1

)

|ψ(t) − ψ

1

(t)| ≤ |ψ(t

0

) − ψ(t

1

)| +

Z

t

t

0

|ψ

′

(s) − ψ

′

1

(s)|ds

≤ ε

1/2

(1 + |t

1

− t

0

|

1/2

).

Hence