Gruber P. Convex and Discrete Geometry

64 Convex Bodies
in contradiction to the minimality of A. Hence the first case cannot hold. Second
case: x ∈ relbd cl conv R(A). Then we may choose u ∈ S d−1 such that x ∈
H(u)∩cl conv R(A) but cl conv R(A) �⊆ H(u).By(5),x −µ(A + ) ∈ cl conv R(A 0 ).
The set A 0 ∈ M is minimal such that x − µ(A + ) ∈ cl conv R(A 0 ). Otherwise
A ∈ M would not be minimal such that x ∈ cl conv R(A). Since cl conv R(A) �⊆
H(u), Proposition (5) shows that the dimension of cl conv R(A 0 ) is smaller than
the dimension of cl conv R(A). IfR(A 0 ) ={o} we have x = µ(A + ) and we are
done. If not, we may repeat the above argument with x − µ(A + ) and A 0 instead of
x and A to show that x − µ(A + ) − µ(A 0+ ) ∈ cl conv R(A 00 ), where the dimension
of cl conv R(A 00 ) is smaller than the dimension of cl conv R(A 0 ), etc. In any case,
after finitely many repetitions we arrive at (6).
Propositions (2) and (6) finally yield (1), concluding the proof of the theorem.
⊓⊔
Remark. The range of a finite non-atomic vector-valued signed measure is actually
a zonoid and vice versa, as shown by Rickert [835]. See also Bolker [141]. Zonoids
are particular convex bodies. See Sect. 7.3 for a definition.
4.4 Pontryagin’s Minimum Principle
The minimum principle, sometimes also called maximum principle, of Pontryagin
and his collaborators Boltyanskiĭ, Gamkrelidze and Mishchenko [813] is a central
result of control theory. It yields information on optimal controls.
In the following we present a version which makes essential use of support
hyperplanes and which rests on convexity arguments, both finite and infinite
dimensional.
For more information on control theory see, e.g. Pontryagin, Boltyanskiĭ,
Gamkrelidze, Mishchenko [813] and Agrachev and Sachkov [2].
The Time Optimal Control Problem
Consider a system of linear differential equations with given initial value:
(1) ˙x = Ax + Bu, x(0) = a,
where A and B are real d × d, respectively, d × c matrices and a ∈ E d .Acontrol
u :[0, +∞) → C is the parametrization of a curve which is contained in a given
convex body C in E c . We suppose that each of its c components is measurable (with
respect to Lebesgue measure on R). Given a control u,byasolution of (1), we mean
a continuous parametrization x :[0, +∞) → E d of a curve, such that the components
of x are almost everywhere differentiable and (1) holds almost everywhere on
[0, +∞). A control u is said to transfer the initial state a to a state b ∈ E d \{a} in
time t if x(t) = b holds for the corresponding solution x of (1).
The time optimal control problem is to find a control u which transfers a to b in
minimum time.