Find the extremal in the following constrained problem, with two differential equation constraints: min J Z

Find the extremal in the following constrained problem, with two differential equation constraints: min J ¼ Z 1 0 1 2  y2 1 þ y2 2 þ y2 3  dt s:t: y0 1 ¼ y2  y1 y0 2 ¼  2y1  3y2 þ y3 y1ð0Þ ¼ 5; y2ð0Þ ¼ 5; y1ð1Þ ¼ free; y2ð1Þ ¼ free

a. Find the necessary conditions to solve the problem setting up the augmented integrand function and the EulereLagrange conditions.

b. Once you have identified the system of the five differential equations, set them up in Excel and using the shooting method, changing the two Lagrange multipliers at t0 ¼ 0 (Solver), find the numerical solutions that optimizes the functional.

c. Plot the solutions in a chart. Hint: the augmented integrand function will be given by: FaðyðtÞ; y0 ðtÞ; lðtÞ; tÞ ¼ 1 2  y2 1 þ y2 2 þ y2 3  þ l1  y2  y1  y0 1  þ l2   2y1  3y2 þ y3  y0 2  Now, we just need to find the differential equations of the system via the following five EulereLagrange conditions: vFa vy1  d dt  vFa vy0 1  ¼ 00l0 1 ¼ y1 þ l1 þ 2l2 vFa vy2  d dt  vFa vy0 2  ¼ 00l0 2 ¼ y2  l1 þ 3l2 518 vFa vy3  d dt  vFa vy0 3  ¼ 00y3 ¼ l2 vFa vl1  d dt  vFa vl0 1  ¼ 00y2  y1  y0 1 ¼ 0 vFa vl2  d dt  vFa vl0 2  ¼ 00  2y1  3y2 þ y3  y0 2 ¼ 0