By Evans L.C.
Those lecture notes construct upon a path Evans taught on the collage of Maryland throughout the fall of 1983.
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Additional resources for An Introduction To Mathematical Optimal Control Theory (lecture notes) (Version 0.1)
We have the constraint 0 ≤ α(t) ≤ 1; that is, A = [0, 1] ⊂ R. The economy evolves according to the dynamics (ODE) x(t) ˙ = α(t)x(t) (0 ≤ t ≤ T ) x(0) = x0 where x0 > 0 and we have set the growth factor k = 1. We want to maximize the total consumption T (P) (1 − α(t))x(t) dt P [α(·)] := 0 51 How can we characterize an optimal control α∗ (·)? Introducing the maximum principle. We apply Pontryagin Maximum Principle, and to simplify notation we will not write the superscripts ∗ for the optimal control, trajectory, etc.
Hence if we know p(·), we can design the optimal control α(·). So next we must solve for the costate p(·). We know from (ADJ) and (T ) that p(t) ˙ = −1 − α(t)[p(t) − 1] p(T ) = 0 52 (0 ≤ t ≤ T ) Since p(T ) = 0, we deduce by continuity that p(t) ≤ 1 for t close to T , t < T . Thus α(t) = 0 for such values of t. Therefore p(t) ˙ = −1, and consequently p(t) = T − t for times t in this interval. So we have that p(t) = T − t so long as p(t) ≤ 1. And this holds for T −1≤t≤T But for times t ≤ T − 1, with t near T − 1, we have α(t) = 1; and so (ADJ) becomes p(t) ˙ = −1 − (p(t) − 1) = −p(t).
Then r is nonincreasing, r(t∗ ) = 0, and consequently r > 0 on [0, t∗ ), r < 0 on (t∗ , τ ]. But (M) says 1 if r(t) < 0 α(t) = 0 if r(t) > 0. Thus an optimal control changes just once from 0 to 1; and so our earlier guess of α(·) does indeed satisfy the Pontryagin Maximum Principle. 5 MAXIMUM PRINCIPLE WITH TRANSVERSALITY CONDITIONS Consider again the dynamics (ODE) ˙ x(t) = f (x(t), α(t)) (t > 0) In this section we discuss another variant problem, one for which the intial position is constrained to lie in a given set X0 ⊂ Rn and the ﬁnal position is also constrained to lie within a given set X1 ⊂ Rn .