Nonlinear Optimal Control Theory

138 Nonlinear Optimal Control Theory

Proof. Write the variational pr oblem as a control problem by writing the

integrand as f

0

(t, x, z), adding the state equation x

′

= z and taking Ω(t, x) =

R

n

for all (t, x). It is readily checked that Assumption 5 .4.1 holds for the

control problem. Since the trajectories of the control problem are the same as

those for the variational problem, the trajecto ries for the control problem lie

in a c ompact set.

In the control formulation,

Q

+

(t, x) = {(y

0

, y) : y

0

≥ f

0

(t, x, z), y = z}

= {y

0

: y

0

≥ f

0

(t, x, y) y ∈ R

n

}.

Since for each (t, x), f

0

(t, x, y) is c onvex in y, it is readily checked that the

sets Q

+

(t, x) are convex. ...