(1) Find the extremal x = x*(t) for the following: L (4.8 (4.82 + ?)dt, x(0) = 0, 2(1) = 1. i.e. satisfying (1) (4.3* (t

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1 Find The Extremal X X T For The Following L 4 8 4 82 Dt X 0 0 2 1 1 I E Satisfying 1 4 3 T 1 (105.18 KiB) Viewed 62 times

(1) Find the extremal x = x*(t) for the following: L (4.8 (4.82 + ?)dt, x(0) = 0, 2(1) = 1. i.e. satisfying (1) (4.3* (t)) = 2*(t). (ü) For the extremal x*(t), you show that (2) S. 14*(0)2 + (x*(? dt = ["la **() zce) + 2*() =(0) dt for all x = x(t) € C2 with x(0) = 0, <(1) = 1. (Hints: Multiplying equation (1) with x*(t) - X(t), integrating in t yields (3) L. 42" () {z*(8) – 2(t)\ dt = = \( x*(0)[**(t) - X(t)] dt. and then integration by parts in (3) with x*(0) - *(0) = 0 and x*(1) – x(1) = 0.) (iii) You prove that < = x*(t) is a minimizing curve. Hints: use inequality ab < }(a+b) for any two real number a, b E R. Then you use the identity (2)