Nonlinear Optimal Control Theory

264 Nonlinear Optimal Control Theory

Hence

t

1

= {−v

0

/d + [(v

0

/d)

2

+ 4(gy

1

/2d

2

)]

1/2

}/(gy

1

/d

2

).

We have just shown that the set of ordinary admissible pairs is not empty

and hence that the set o f relaxed admissible pairs is not empty. Since inf{t

f

: t

f

terminal time o f an admissible trajectory} is les s than or equal to t

1

, it follows

that we can restrict our a ttention to admissible pairs defined on compact

intervals [0, t

f

] ⊆ [0, t

1

]. If we set I = [0, t

1

], we need only consider

b

f on

I × R

2

.

It is easy to show tha t there exists a constant K such that

|h(ξ, η), f(τ, ξ, η)i| ≤ K(|ξ|

2

+ |η|

2

+ 1)

for all t in I, and all (ξ, η) in R

2

and all −π ≤ z ≤ π. Hence ...