Evans L.C.'s An Introduction To Mathematical Optimal Control Theory PDF

By Evans L.C.

Those lecture notes construct upon a direction Evans taught on the collage of Maryland through the fall of 1983.

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Additional info for An Introduction To Mathematical Optimal Control Theory (lecture notes) (Version 0.1)

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So that x0 ∈ C(tn ) and tn → τ ∗ . Since x0 ∈ C(tn ), there exists a control αn (·) ∈ A such that tn x0 = − X−1 (s)N αn (s) ds. 0 If necessary, redefine αn (s) to be 0 for tn ≤ s. By Alaoglu’s Theorem, there exists a subsequence nk → ∞ and a control α∗ (·) so that αn ∗ 31 α∗ . We assert that α∗ (·) is an optimal control. It is easy to check that α∗ (s) = 0, s ≥ τ ∗ . Also tnk x =− 0 X −1 t1 (s)N αnk (s) ds = − 0 X−1 (s)N αnk (s) ds, 0 since αnk = 0 for s ≥ tnk . Let nk → ∞: t1 x =− 0 X −1 τ∗ ∗ (s)N α (s) ds = − 0 X−1 (s)N α∗ (s) ds 0 because α∗ (s) = 0 for s ≥ τ ∗ .

I) If we have T > 0 fixed and T r(x(t), α(t)) dt + g(x(T )), P [α(·)] = 0 then (T) says p∗ (T ) = ∇g(x∗ (T )), 61 in agreement with our earlier form of the terminal/transversality condition. (ii) Suppose that the surface X1 is the graph X1 = {x | gk (x) = 0, k = 1, . . , l}. Then (T) says that p∗ (τ ∗ ) belongs to the “orthogonal complement” of the subspace T1 . But orthogonal complement of T1 is the span of ∇gk (x1 ) (k = 1, . . , l). Thus l p∗ (τ ∗ ) = λk ∇gk (x1 ) k=1 for some unknown constants λ1 , .

Next is a simple model for the trading of a commodity, say wheat. We let T be the fixed length of trading period, and introduce the variables x1 (t) = money on hand at time t x2 (t) = amount of wheat owned at time t α(t) = rate of buying or selling of wheat q(t) = price of wheat at time t (known) λ = cost of storing a unit amount of wheat for a unit of time. We suppose that the price of wheat q(t) is known for the entire trading period 0 ≤ t ≤ T (although this is probably unrealistic in practice).

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An Introduction To Mathematical Optimal Control Theory (lecture notes) (Version 0.1) by Evans L.C.


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