Assignment 2 on Optimal Control

Assignment 2 on Optimal Control - Optimal Control...

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Optimal Control University of Florida Anil V. Rao Question 1 Using the approach for calculus of variations developed in class (not the approach used in Kirk’s book), derive the necessary conditions for optimality that minimize the integral J = integraldisplay t f t 0 L [ x ( t ) , ˙ x ( t ) , t ] dt for the following sets of boundary conditions: t 0 fixed, x ( t 0 )= x 0 fixed; t f fixed, x ( t f )= x f fixed t 0 fixed, x ( t 0 )= x 0 fixed; t f fixed, x ( t f )= x f free t 0 fixed, x ( t 0 )= x 0 fixed; t f free, x ( t f )= x f fixed t 0 fixed, x ( t 0 )= x 0 fixed; t f free, x ( t f )= x f free Question 2 Prove the following theorem (known as the fundamental lemma of variational calculus ). Given a continuous function f ( t ) on the time interval t [ t 0 , t f ] and that integraldisplay t f t 0 f ( t ) δx ( t )=0 for all continuous functions δx ( t ) on t [ t 0 , t f ] such that δx ( t 0 )= δx ( t f )=0 . Then δx ( t ) must be identically zero on t [ t 0 , t f ] . Question 3 (Kirk 1998) Determine the extremal solutions of the following two functionals: (1) J ( x ( t ))= integraltext 1 0 [ x 2 ( t ) - ˙ x 2 ( t )] dt , x (0)=1 , x (1)=1 (2) J ( x ( t ))=
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