Nonlinear Optimal Control Theory

154 Nonlinear Optimal Control Theory

We now make the further assumption that the constraint mapping Ω is

sufficiently smooth so that for every vector z ∈ Ω(τ) there exists a continuous

function v de fined on some interval [τ, τ + ∆t], ∆t > 0, with v(τ) = z and

v(s) ∈ Ω(s) on [τ, τ + ∆t]. In particular , if Ω is a constant mapping, that

is, Ω(t) = C for all t, then we may take v(s) = z on [τ, τ + ∆t]. Under the

assumption just made concerning Ω, we can combine (6.2.10) and (6.2.11) to

get the relation

W

τ

(τ, ξ) = max

zεΩ(τ)

[−f

0

(τ, ξ, z) − hW

ξ

(τ, ξ), f (τ, ξ, z)i], (6.2.12)

with the maximum being attained at z = U (τ, ξ). Equatio n (6.2.12) is so me-

times