Nonlinear Optimal Control Theory

Hamilton-Jacobi Theory 357

Lemma 12.6.6. Let V be a locally Lipschitz continuous function that satisfies

(12.6.2). Then there exists a positive number M, an integer i

0

> 0, and a

sequence {δ

i

}

∞

i=1

, of positive numbers such that for all i ≥ i

0

0 < δ

i

< i

−1

the following holds. Corresponding t o each integer i ≥ i

0

and point (τ, ξ) ∈

[0, T −i

−1

] ×MB there exists a number δ ∈ (δ

i

, i

−1

) and a point z in C such

that

δ

−1

(V (τ + δ, ψ(τ + δ; τ, ξ, v

z

)) − V (τ, ξ)) < i

−1

. (12.6.9)

Proof. Le t i

0

be as in Lemma 12.6.5. Choose any i ≥ i

0

and let (τ, ξ) ∈

[0, T ] ×M B. The i chosen will be fixed for the remainder of this proof. Then

since V satisfies (12.6.2), there exists ...