Nonlinear Optimal Control Theory

The Maximum Principle and Some of Its Applications 159

of (∂T /∂σ

i

, ∂X/∂σ

i

), Eq . (6.2.30) states that at (τ

1

, ξ

1

) the vector (W

t

−

g

t

, W

x

−g

x

) is either zero or is orthogonal to Γ

i

. We assume that orthogonality

holds. From (6.2.31) and (6 .2.32) we see that this statement is equivalent to

the sta tement that (H − g

t

, W

x

− g

x

) is o rthogonal to Γ

i

at (τ

1

, ξ

1

) a nd to

the statement that (H − g

t

, −λ − g

x

) is orthogonal to Γ

i

. Since this is true

for each coordinate curve Γ

i

, i = 1, . . . , n and since the tangent vectors to

the Γ

i

at (τ

1

, ξ

1

) ge ne rate the tangent plane to T at (τ

1

, ξ

1

), the following

statement is true. The vector (W

t

− g

t

, W

x

− g

x

), or equivalently ...