UBC Mathematics 402(201)—Assignment 11
Due at 09:00 on Thursday 02 April 2015
1. Consider the free-right-endpoint problem
min
b>a, x
(Z
b
a
)
p
f (t, x(t)) 1 + ẋ2 (t) dt : x(a) = A, x(b) = g(b) ,
where f > 0 and g are given C 1 functions. Use the transversality condition to show that any local
solution to this problem must meet the graph of g orthogonally.
2. Consider the set of piecewise smooth functions x(t) > 0 starting from the point x(0) = 1 and
terminating at their first contact time b > 0 with the straight line x = t − 5. Find the only arc in
this set that could possibly minimize
Z
0
b
p
1 + ẋ(t)2
dt.
x(t)
3. Consider the set of piecewise smooth functions starting from the point x(1) = 9/8 and terminating
at their first contact time b > 1 with the curve x = 2 + (t − 1)2 . In this set there is only one arc
that can possibly minimize
Z b
ẋ(t)2
dt.
t3
1
Find this arc. (Clue: The equation determining the final time may look daunting, but a search for
integer solutions will not be disappointing.)
4. In the general theory of convex functions, the value +∞ is allowed. Try this out: using the sensible
convention that λ(+∞) + (1 − λ)(+∞) = +∞ for all λ ∈ [0, 1], prove that the function below is
convex, and evaluate f ∗ (p):
(
x ln x, if x > 0,
f (x) := 0,
if x = 0,
+∞,
if x < 0.
p
Then do the same for g(x) = ex and h(x) = 1 + x2 . (Note: Both g∗ (p) and h∗ (p) take on the
value +∞ for some range of p-values.)
5. The arc x
b(t) = et is suspected of solving the following problem:
min
Z
1
2
2
ẋ(t) + x(t)
0
dt : x(0) = 1, x(1) = e .
Find two essentially different verification functions that confirm this fact. [Clue: For certain functions f and g, chosen with x
b in mind, there is a verification function of the form W (t, x) =
f (t)x + g(t). It works because the integrand is convex.]
File “hw11”, version of 26 Mar 2015, page 1.
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