An Introduction To Mathematical Optimal Control Theory by Evans L.C.

By Evans L.C.

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

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We claim that α1 (·), α2 (·) ∈ K. To see this, observe that − t 0 X−1 (s)N α1 (s) ds = − t 0 X−1 (s)N α∗ (s) ds − ε t 0 X−1 (s)N β(s) ds X−1 (s)N β(s) ds = x0 . = x0 − ε F IF (β(·))=0 Note also α1 (·) ∈ A. Indeed, α1 (s) = α∗ (s) (s ∈ / F) ∗ α1 (s) = α (s) + εβ(s) (s ∈ F ). But on the set F , we have |α∗i (s)| ≤ 1 − ε, and therefore |α1 (s)| ≤ |α∗ (s)| + ε|β(s)| ≤ 1 − ε + ε = 1. Similar considerations apply for α2 . Hence α1 , α2 ∈ K, as claimed above. 3. Finally, observe that α1 = α∗ + εβ, α1 = α∗ α2 = α∗ − εβ, α2 = α∗ .

Then τ∗ T h X −1 τ∗ (s)N α(s) ds ≤ 0 hT X−1 (s)N α∗ (s) ds; 0 and therefore τ∗ hT X−1 (s)N (α∗ (s) − α(s)) ds ≥ 0 0 for all controls α(·) ∈ A. 3. We claim now that the foregoing implies hT X−1 (s)N α∗ (s) = max{hT X−1 (s)N a} a∈A for almost every time s. For suppose not; then there would exist a subset E ⊂ [0, τ ∗ ] of positive measure, such that hT X−1 (s)N α∗ (s) < max{hT X−1 (s)N a} a∈A ˆ as follows: for s ∈ E. Design a new control α(·) ˆ α(s) = α∗ (s) (s ∈ / E) α(s) (s ∈ E) where α(s) is selected so that max{hT X−1 (s)N a} = hT X−1 (s)N α(s).

0 −1 = −I and consequently t2 2 M + ... 2! t2 t3 t4 = I + tM − I − M + I + . . 2! 3! 4! 2 4 t t t3 t5 = (1 − + − . . )I + (t − + − . . )M 2! 4! 3! 5! cos t sin t = cos tI + sin tM = . − sin t cos t etM = I + tM + So we have X−1 (t) = and X−1 (t)N = cos t sin t whence hT X−1 (t)N = (h1 , h2 ) cos t sin t − sin t cos t − sin t cos t − sin t cos t 0 1 = − sin t cos t ; = −h1 sin t + h2 cos t. According to condition (M), for each time t we have (−h1 sin t + h2 cos t)α∗ (t) = max {(−h1 sin t + h2 cos t)a}.

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