Nonlinear Optimal Control Theory

342 Nonlinear Optimal Control Theory

Proof. To simplify notation we suppress the dependence on τ a nd write ψ(·, x)

for ψ(·, τ, x) and ψ(·, x

′

) for ψ(·, τ, x

′

). From (12.2.1) and Gronwalls’ Lemma

we get that for τ ≤ t ≤ T

|ψ(t, x) − ψ(t, x

′

)| ≤ |x −x

′

|+

Z

t

τ

|(f(s, ψ(s, x), µ

s

) − f(s, ψ(s, x

′

), µ

s

)|ds

≤ |x − x

′

|+

Z

t

τ

K(s)|ψ(s, x) −ψ(s, x

′

)|ds

Thus,

|ψ(t, x) − ψ(t, x

′

)| ≤ |x −x

′

|exp

Z

t

τ

K(s)ds ≤ |x − x

′

|A,

where A = exp

R

I

K(s)ds.

We return to the proof of the theorem. Let (τ, ξ) be a point in R

0

and let

(ψ, µ) = (ψ(·, τ, ξ, µ), µ(τ, ξ)) be an optimal relaxed pair for Problem 12.2.1.

Since the problem is in Mayer form

W (τ, ξ) = g(T, ψ(T, τ, ξ)) = g(T, ψ(T )). (12.3.5) ...