“…Proof: To ensure (3) is satisfied, we must show the optimal control solution (26)-(30) in Proposition 1 can ensure (3) being satisfied. For Cases 1, 2 and 4, from their optimal control solutions (26), (27) and (30), it is easy to see that (3) is satisfied. For Cases 3 and 5, from their optimal control solutions (31) or (32) and (28) or (29), if F w d,t = W t − S w,t < D max , E w t = 0 and F s,t ≥ 0; If F w d,t = D max , we have F w s,t = 0 and E w t ≥ 0.…”