Nonlinear Optimal Control Theory

Hamilton-Jacobi Theory 347

R

0

, |(t, x) − (τ, ξ)| < ε}. Since ψ is continuous, given an ε

′

> 0, there exists

a δ(ε

′

) with 0 < δ(ε

′

) < ε

′

such that if τ ≤ s ≤ τ + δ, then

ψ

′

(s) ∈ Q(N

ε

(τ, ξ)) a.e., (12.4.6)

where ε = δ(ε

′

). Let K

ε

denote the set of points in [τ, τ + δ] a t which the

inclusion (12.4.6) holds. Then the Lebesgue mea sure of K

ε

equals δ. Thus,

Z

τ +δ

τ

ψ

′

(s)ds = δ

Z

τ +δ

τ

ψ

′

(s)



ds

δ



= δ

Z

K

ε

ψ

′

(s)



ds

δ



.

From (12 .4.6) we g et that

cl co {ψ

′

(s): s ∈ K

ε

} ⊆ cl co {Q(N

ε

(τ, ξ))}.

From Lemma 3.2.9 we get that

cl co {ψ

′

(s): s ∈ K

ε

} = cl



Z

K

ε

ψ

′

(s)dµ: µ ∈ P(K

δ

)



,

where P(K

δ

) is the set of probability measures on K

δ

. Hence

Z

K

ε

ψ

′

(s)



ds

δ



∈ cl co {Q(N

ε

(τ, ξ))}.

Let

h

δ

≡

Z

τ