# hw10sol - M402(201) SolutionsAssignment 10 1. (a) Suppose a...

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M402(201)Solutions—Assignment101.(a) Suppose a minimizing arc exists; call itx.Then there must be constantsλ0∈ {0,1}andλR,not both zero,such thatxis extremal fortildewideL(t, x, v)
2UBC M402 Solutions #10Herek=1 works by inspection, givingr=2 and the admissible extremalx(t) =1 +radicalbig2(t1)2.To prove that this is a true minimizer, note that the map (x, v)mapsto→tildewideL(x, v) is convex. Thusxminimizes Λ +λΓ subject to the given endpoint conditions, and consequentlyxminimizes Λsubject to the constraint Γ =γ.2.(a) Here the augmented Lagrangian istildewideL(x, v) =λ0(v2x2) +λ(v2+x2). Settingλ0= 0 (henceλnegationslash= 0) givestildewideL=λ(v2+x2). ThustildewideLvv= 2λhas the same (nonzero) sign for all (t, x, v) andconsequently any extremal must be aC2solution of (DEL):ddt(2 ˙x) = 2x=¨xx= 0 =x(t) =Aet+Bet.HereAandB

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Term
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