d) For any u E Mad, define a process Ct(u) = (QX? + Ru, )dr + V(t, X+), +20. Using the HJB equation to prove that C. (u) is a submartingale in general, and is a martingale when u is optimal. e) Find the optimal feedback control in terms of the value function (by finding the minimizer in HJB equation). f) Assume that V is quadratic, namely V(s, x) = P(t)x2 + (t)x + f(t), where P(.), 4(.), and f(.) are continuously differentiable functions, derive the differential equations and other necessary conditions that functions P(.), 4(.), and f(.) must satisfy, and derive the optimal control u*. g) Assume that A = 1/2, B = 2, C = E = F = P = R = 1, D = Q =0. Calculate the optimal control