# Structure of Approximate Solutions of Optimal Control Problems

## Alexander J. Zaslavski

Language: English

Pages: 133

ISBN: B00EBX3OY6

Format: PDF / Kindle (mobi) / ePub

This titleexamines the structure of approximate solutions of optimal control problems considered on subintervals of a real line. Specifically at the properties of approximate solutions which are independent of the length of the interval. The results illustrated in this book look into the so-called turnpike property of optimal control problems. The author generalizes theresultsof the turnpike property by considering a class of optimal control problems which is identified with the corresponding complete metric space of objective functions.This establishes the turnpike property for any element in a set that is ina countable intersectionwhich is open everywhere dense sets in the space of integrands; meaning that the turnpike property holds for most optimal control problems. Mathematicians working in optimal control and the calculus of variations and graduate students will find this bookuseful and valuable due to its presentation of solutions to a number of difficult problems in optimal controland presentation of new approaches, techniques and methods.

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t1 + Δ1 ] : |x(t) − x∗ (t)| ≤ δ}, t˜2 = inf{t ∈ [t1 + Δ1 , T2 ] : |x(t) − x∗ (t)| ≤ δ}. (2.140) 48 2 Turnpike Properties of Optimal Control Problems Then t˜1 < t1 + Δ1 , t˜2 > t1 + Δ1 + Δ2 , |x(t˜i ) − x∗ (t˜i )| = δ, i = 1, 2. There exist b1 ∈ (t˜1 , t1 + Δ1 ) and b2 ∈ (t1 + Δ1 + Δ2 , t˜2 ) (2.141) |x(bi ) − x∗ (bi )| < γ, i = 1, 2. (2.142) |x(t) − x∗ (t)| > δ for all t ∈ [b1 , b2 ]. (2.143) such that It is easy to see that By (2.141)–(2.143) and the choice of U1 , Δ2 [see (2.133),

respectively. It follows from our construction that r(k) = p, Sk = tp and that (C1 and (C2) hold. 2.4 Auxiliary Results 67 By (2.220) and (C1), S ≥ I g (T1 , T2 , x, u) − inf{σ g (T1 , T2 , y) : (T1 , y) ∈ A} ≥ {I g (Si , Si+1 , x, u) − U f (Si , Si+1 , x(Si ), x(Si+1 )) : i is an integer such that 1 ≤ i < k − 1} ≥ (k − 2)δ0 and k ≤ 2 + δ0−1 S. (2.240) Set A = {i ∈ {1, . . . , k} : i < k and Si+1 − Si > 2Δ2 + 2Δ1 }. (2.241) i ∈ A. (2.242) tr(i+1) − tr(i) = Si+1 − Si > 2Δ2 + 2Δ1 .

trajectorycontrol pair x1 : [T, T + τ ] → Rn , u1 : [T, T + τ ] → Rm which satisfies x1 (T1 ) = y1 , x1 (T2 ) = z1 , I f (T1 , T2 , x1 , u1 ) = U f (T1 , T2 , y1 , z1 ) there exists a trajectory-control pair x2 : [T, T + τ ] → Rn , u2 : [T, T + τ ] → Rm such that x2 (T1 ) = y2 , x2 (T2 ) = z2 , |I f (T1 , T2 , x2 , u2 ) − I f (T1 , T2 , x1 , u1 )| ≤ , |x1 (t) − x2 (t)| ≤ , t ∈ [T1 , T2 ]. Proof. Set M0 = sup{|U f (T, T + τ, y, z)| : T ∈ R1 , y, z ∈ Rn , |y|, |z| ≤ M + 4} < ∞. (4.35) By

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following inequality holds: Assume that T1 ∈ R1 , T2 ≥ T1 + 2b1 and that a trajectory-control pair x : [T1 , T2 ] → Rn , u : [T1 , T2 ] → Rm satisfies conditions (a) and (b). Let τ1 and τ2 be as guaranteed in condition (b). We show that (2.41) holds. Assume the contrary. Then there exists T0 ∈ [T1 , T2 ] such that |x(T0 )| > S1 . (2.48) By condition (b) there exists a trajectory-control pair x1 : [T1 , T2 ] → Rn , u1 : [T1 , T2 ] → Rm such that x1 (Tj ) = x(Tj ), x1 (τj ) = x∗ (τj ), j = 1, 2,