By Evans L.C.

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

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

**Example text**

We deduce therefore that α∗ switches from +1 to −1, and vice versa, every π units of time. Geometric interpretation. Next, we ﬁgure out the geometric consequences. When α ≡ 1, our (ODE) becomes x˙ 1 = x2 x˙ 2 = −x1 + 1. 38 r1 (1,0) In this case, we can calculate that d ((x1 (t) − 1)2 + (x2 )2 (t)) = 2(x1 (t) − 1)x˙ 1 (t) + 2x2 (t)x˙ 2 (t) dt = 2(x1 (t) − 1)x2 (t) + 2x2 (t)(−x1 (t) + 1) = 0. Consequently, the motion satisﬁes (x1 (t) − 1)2 + (x2 )2 (t) ≡ r12 , for some radius r1 , and therefore the trajectory lies on a circle with center (1, 0), as illustrated.

Since x(·) solves (ODE), we have x1 ∈ K(t, x0 ) if and only if t x1 = X(t)x0 + X(t) X−1 (s)N α(s) ds = x(t) 0 for some control α(·) ∈ A. 2 (GEOMETRY OF THE SET K). The set K(t, x0 ) is convex and closed. Proof. 1. (Convexity) Let x1 , x2 ∈ K(t, x0 ). Then there exists α1 , α2 ∈ A such that t x1 = X(t)x0 + X(t) X−1 (s)N α1 (s) ds 0 t x2 = X(t)x0 + X(t) X−1 (s)N α2 (s) ds. 0 Let 0 ≤ λ ≤ 1. Then t λx1 + (1 − λ)x2 = X(t)x0 + X(t) X−1 (s)N (λα1 (s) + (1 − λ)α2 (s)) ds, 0 32 ∈A and hence λx1 + (1 − λ)x2 ∈ K(t, x0 ).

2, 6, . . = 3, 7, . . ; 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}.