Gruber P. Convex and Discrete Geometry

i.e.
i.e.
�T
0
4 Support and Separation 67
B T e AT (T −s) w · � u(s) − v(s) � ds ≥ 0forv ∈ C(T ),
�T
0
B T y(s) · � u(s) − v(s) � ds ≥ 0forv ∈ C(T ).
Since the latter inequality holds for all v ∈ C(T ), we see that for w(s) = B T y(s)
(�= o),
(10) w(s) · � u(s) − v(s) � ≥ 0 for each v ∈ C(T ) and almost every s ∈[0, T ].
Otherwise, there is a control v ∈ C(T ) such that the inner product in (10) is negative
on a subset of the interval [0, T ] of positive measure. Clearly, for the control in C(T )
which coincides with this v on the subset and with u outside of it, the integral over
the expression in (10) is negative. This is the required contradiction. (10) implies that
i.e.
i.e.
w(s) · u(s) ≥ w(s) · v(s) for each v ∈ C(T ) and almost every s ∈[0, T ],
w(s) · u(s) = max{w(s) · v : v ∈ C} for almost every s ∈[0, T ],
� �
u(s) ∈ C ∩ HC w(s) for almost every s ∈[0, T ]. ⊓⊔
A Geometric Interpretation of the Minimum Principle and Bang-Bang
Controls
The above version of the minimum principle says that, for the time optimal control
u, the following statement � �holds: for almost every t the point � u(t) � is contained in the
support set C ∩ HC w(t) as the support hyperplane HC w(t) rolls continuously
over C. IfC is strictly convex, each support set consists of a single point. The time
optimal control u then may be chosen as a continuous parametrization of a curve
on bd C. Suppose now that C is the unit cube in Ed � � and such that the support set
C ∩ HC w(s) is a vertex of C for all times t ∈[0, T ], with a finite set of exceptions.
Then there are times 0 = t0 < t1 < ··· < tn = T , vertices v0,...,vn of C and a
time optimal control u such that
u(t) = vk for t ∈ (tk−1, tk), k = 1,...,n.
Then u is said to be a bang-bang control. Essentially this says, if each of the admissible
strategies of a control problem varies independently in a given interval, time
optimal controls consist of a finite sequence of pure strategies.