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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. Typeset at 21:28 March 26, 2015.