Assignment 2 on Optimal Control

Assignment 2 on Optimal Control - Optimal Control...

This preview shows pages 1–2. Sign up to view the full content.

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 ))=

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern