Further Exercises on the Euler-Lagrange Equation

[Hint: Find S t in terms of x t and integrate the debt equation.]9. Maximise R 1f(t)p x(t) dt subject to R 1x(t) dt = c.010. Maximise R 10 f(t)x(t) dt subject to R 1x(t) dt = c where 1 .0011. For the problem:subject tominZ T0(x 2 + _x 2 ) dtx(0) = c; x(T ) = 0;the trajectory is known to be of the formshow thatx T (t) = Ae t + Be t ;lim A T = 0:T !1Show also that, pointwise lim T !1 x T (t) = ce t . Interpreting x T (t) = x T (T ) fort > T , verify that the limit is uniform.12. For the problem:min 1 2subject to x(0) = 1; x(1) = 0 andZ 10Z 10(x 2 + _x 2 ) dtxe 2t dt = 1show that the optimal curve is x = 9e t 8e 2t .13. For the problem:minZ 20f r 2 + 2 _r 2 g 1=2 d