Nonlinear Optimal Control Theory

362 Nonlinear Optimal Control Theory

Proof. Under Assumption 12.8.1 the relaxed control problem has a solution

for each initial point (t, x) in R

0

and the value function is Lipschitz continuous

on compact s ubs ets of R

0

. (See Theorem 4.3.5 and Theorem 12.3.2.)

Let z be an arbitrary element of

e

C. Let ψ( ; τ, ξ, z) denote the trajectory

with initial point (τ, ξ) and v(t) = z. It follows from (12.2.2) and Corol-

lary 4.3.15 that there exists a constant A > 0 such that for all (τ, ξ) in R

OL

and all z in

e

C, |ψ(t; τ, ξ, z)| ≤ A for all τ ≤ t ≤ T . The function f is uniformly

continuous on [0, T ] × B

A

×

e

C where B

A

is the closed ball in R

n

of radius

A with center ...