Nonlinear Optimal Control Theory

352 Nonlinear Optimal Control Theory

Then

V (t + δ

k

, x + δ

k

h) − V (t, x) = {V (t + δ

k

, x + δ

k

h) − V (t

k

+ δ

k

, x

k

+ δ

k

h)}

+ {V (t

k

+ δ

k

, x

k

+ δ

k

h) − V (t

k

, x

k

)} + {V (t

k

, x

k

) − V (t, x)}

≡ A

k

+ B

k

+ C

k

From the Lipschitz continuity of V and (12.5.5) we have that

(A

k

+ C

k

) = O(|t − t

k

|) + O(|x −x

k

|) = O(δ

2

k

)

Hence

lim

k→∞

(A

k

+ C

k

)δ

−1

k

= 0. (12.5.6)

Since V is differentiable at (t

k

, x

k

),

B

k

= [V

t

(t

k

, x

k

) + hV

x

(t

k

, x

k

), hi]δ

k

+ o(δ

k

(1 + h))

≥ [V

t

(t

k

, x

k

) + inf

h∈Q(t,x)

hV

x

(t

k

, x

k

), hi]δ

k

+ o(δ

k

).

Hence

B

k

≥ −[−V

t

(t

k

, x

k

) + sup

h∈Q(t,x)

h−V

x

(t

k

, x

k

), hi]δ

k

+ o(δ

k

).

At points of differentiability a viscosity solution satisfies (12.5.1) in the ordi-

nary sense. Hence

lim inf

k→∞

B

k

δ

−1

k

≥ 0. (12.5.7)

From (12 .5.4), (12.5.6), and (12.5.7) we get that

D

−

V (t, x; 1, h) = lim

k→∞

(A

k

+ B

k

+ C

k

)