QE2013_AC-3_ECE580-4

Part 1,2,3,4,5

(i)
Solution:
$ x(3) = x(2) + 2u(2) = x(1) + 2u(1) + 2u(2) = x(0) + 2u(0) + 2u(1) + 2u(2) $

$ \because x(0) = 3, x(3) = 9 $

$ \therefore \ Constraint: 2u(0) + 2u(1) + 2u(2) = 6 $

Let $ f(u) = u^2(0) + u^2(1) + u^2(2), h(u) = 2u(0) + 2u(1) + 2u(2) - 6 $

Let u* be a local minimizer. Lagrange theorem says there exists a λ such that:

$ \nabla f(u*) + \lambda \nabla h(u*) = 0 \\ h(u*) = 0 $ Back to QE2013 AC-3 ECE580