Gruber P. Convex and Discrete Geometry

66 Convex Bodies
(6) R(t) ⊆ E d is compact and convex for t ≥ 0.
The second property of R(t), that will be used below, is a sort of weak right-hand
side continuity:
(7) Let t1 ≥ t2 ≥···→ T and b ∈ R(tn) for n = 1, 2,... Then b ∈ R(T ).
By definition,
b = e Atn
�tn
a + e A(tn−s)
Bun(s) ds for suitable un ∈ C(tn)
0
T
�
= e AT �
a +
0
e A(T −s) �
Bun(s) ds
��eAtn AT
+ − e � �tn
a + e A(tn−s)
�
Bun(s) ds +
T
0
T
� A(tn−s) A(T −s)
e − e � �
Bun(s) ds .
The quantity in the first bracket is contained in R(T ) and R(T ) is compact by (6).
The quantity in the second bracket tends to o as n →∞. Hence b ∈ R(T ). The third
required property of R(t) is as follows:
(8) If b ∈ int R(T ), then b ∈ R(t) for all t ≤ T sufficiently close to T .
To see this, choose a simplex in int R(T ) with vertices x1,...,xd+1, say, which contains
b in its interior. By the definition of R(T ), there are solutions x1(·), . . . , xd+1(·)
of (1), corresponding to suitable controls in C(T ) which transfer a to x1,...,xd+1 in
time T . That is, x1(T ) = x1,...,xd+1(T ) = xd+1. These solutions are continuous.
Hence, for all t < T sufficiently close to T , the points x1(t),...,xd+1(t) are the
vertices of a simplex in the convex set R(T ) and this simplex still contains b. Hence
b ∈ R(t) by (6).
After these preparations, the proof of the theorem is rather easy. Let T (≥ 0) be
the infimum of all t > 0 such that b ∈ R(t). By(7),b ∈ R(T ), concluding the
proof of statement (i). Since b �= a, a consequence of (i) is that T > 0. To show
(ii), note first that b �∈ int R(T ). Otherwise b ∈ R(t) for suitable t < T by (8), in
contradiction to the definition of T . Hence b is a boundary point of the convex body
R(T ) (see (6)). Theorem 4.1 then shows that
(9) (z − b) · w ≤ 0forz ∈ R(T ),
with a suitable w ∈ E d \{o}. Ifu ∈ C(T ) is a control which transfers a to b in time
T , Propositions (3) and (9) yield the following:
�T
0
e A(T −s) B � v(s) − u(s) � ds · w ≤ 0forv ∈ C(T ),