Exercise 7.15. Consider the discounted infinite-horizon problem, with f (t, x (t), y (t)) = exp (t)

Exercise 7.15. Consider the discounted infinite-horizon problem, with f (t, x (t), y (t)) = exp (−ρt) f (x (t), y (t)), and g (t, x (t), y (t)) = g (x (t), y (t)). Prove that if an admissible pair (ˆx (t), yˆ(t))t≥0 is optimal starting at t = 0 with initial condition x (0) = x0, then (ˆx (t), yˆ(t))t≥s is also admissible and optimal for the problem starting at t = s with initial condition x (s) = x0.