Gruber P. Convex and Discrete Geometry

The Minimum Principle
4 Support and Separation 65
The following result of Pontryagin et al. [813] sheds light on the time optimal control
problem.
Theorem 4.6. Assume that there is a control for (1) which transfers a to b. Then the
following propositions hold:
(i) There is a time optimal control which transfers a to b.
(ii) Let u be a time optimal control which transfers a to b in minimum time T , say.
Then there is a point w ∈ Ed such that for the solution of the initial value problem
(2) ˙y =−AT y, y(T ) = w,
the following statement holds, where w(t) = BT y(t):
� �
u(t) ∈ C ∩ HC w(t) for almost every t ∈[0, T ].
For the geometric meaning of Proposition (ii), see the remarks after the proof of
the theorem.
Proof. Before beginning the proof we collect some tools.
(3) The solutions of (1) and (2) can be presented as follows:
x(t) = e At �t
a + e A(t−s) Bu(s) ds for t ≥ 0,
0
y(t) = e −AT (t−T ) w(�= o) for t ≥ 0.
Here e At is the non-singular d × d matrix I + 1 1! At + 1 2! A2 t 2 + ···. The integral
is to be understood component-wise. For t ≥ 0, consider the Hilbert space
L 2 = L 2 ([0, t], E c ) of all vector-valued functions u :[0, t] →E c with measurable
component functions such that u 2 is integrable on [0, t]. Besides the ordinary topology
on L 2 , there is also the so-called weak topology. With respect to this topology
the following statements hold.
(4) C(t) ={u ∈ L 2 : u(s) ∈ C for s ∈[0, t]} ⊆ L 2
is compact and, trivially, convex for t ≥ 0.
(5) The mapping u → e At �t
a + e A(t−s) Bu(s) ds
0
is a continuous and, trivially, affine mapping of L 2 into E d for t ≥ 0.
Next, we define and investigate the set R(t) reachable in time t,
�
R(t) = e At �
a +
0
t
e A(t−s) �
Bu(s) ds : u ∈ C(t) ⊆ E d , t ≥ 0.
Note that a continuous affine image of a compact convex set is again compact and
convex. Hence Propositions (4) and (5) yield the following property of R(t):