Introduction to Modern Economic Growth is the fact that the value of the functional in (7.28) may not be finite We will deal with some of these issues below The main theorem for the infinite-horizon optimal control problem is the following more general version of the Maximum Principle Before stating this theorem, let us recall that the Hamiltonian is defined by (7.12), with the only difference that the horizon is now infinite In addition, let us define the value function, which is the analogue of the value function in discrete time dynamic programming introduced in the previous chapter: V (t0 , x0 ) ≡ (7.31) max x(t)∈R,y(t)∈R Z ∞ f (t, x (t) , y (t)) dt t0 subject to x˙ (t) = g (t, x (t) , y (t)) , x (t0 ) = x0 and lim x (t) ≥ x1 t→∞ In words, V (t0 , x0 ) gives the optimal value of the dynamic maximization problem starting at time t0 with state variable x0 Clearly, we have that (7.32) V (t0 , x0 ) ≥ Z ∞ f (t, x (t) , y (t)) dt for any admissible pair (x (t) , y (t)) t0 Note that as in the previous chapter, there are issues related to whether the “max” is reached When it is not reached, we should be using “sup” instead However, recall that we have assumed that all admissible pairs give finite value, so that V (t0 , x0 ) < ∞, and our focus throughout will be on admissible pairs (ˆ x (t) , yˆ (t)) that are optimal solutions to (7.28) subject to (7.29) and (7.30), and thus reach the value V (t0 , x0 ) Our first result is a weaker version of the Principle of Optimality, which we encountered in the context of discrete time dynamic programming in the previous chapter: Lemma 7.1 (Principle of Optimality) Suppose that the pair (ˆ x (t) , yˆ (t)) is an optimal solution to (7.28) subject to (7.29) and (7.30), i.e., it reaches the maximum value V (t0 , x0 ) Then, (7.33) V (t0 , x0 ) = Z t1 t0 f (t, xˆ (t) , yˆ (t)) dt + V (t1 , xˆ (t1 )) for all t1 ≥ t0 332