1. The optimal control final representation 詹雲如 0016C026 蔡宗銘 0016C020 丘濟恆 0016C022 1

2. Outline  Joseph-Louis Lagrange  Euler-Lagrange Equation’s Introduction  Euler-Lagrange Equation  Lagrange Multiplier  Example  Assignments  Source 2

3. Joseph-Louis Lagrange  French nationality Italian-American mathematician and astronomer.  French Academy of Sciences.  Famous achievements: (1) Analytical mechanics (2) Celestial mechanics (3) Mathematical analysis (4) Number theory  Contribution 3

4. Euler-Lagrange Equation’s Introduction  Second-order partial differential equation.  It is developed by Leonhard Euler & Joseph-Louis Lagrange in the 1750s in connection with their studies of the tautochrone problem.  Useful for solving optimization problems.  Satisfied by a function of a real argument which is a stationary point of the functional.  Functional is function of function. 4

5. Euler-Lagrange Equation 5

6. Lagrange Multiplier 6

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8. Example 8

9. Review 9

10. Assignments  詹雲如 (1) Joseph-Louis Lagrange. (2) Summarization.  蔡宗銘 (1) Lagrange Multiplier. (2) Example.  丘濟恆 (1) Euler-Lagrange Equation. 10

11. Source  Wikipedia, Joseph-Louis Lagrange https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange  Wikipedia, Euler-Lagrange Equation https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_e quation https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_e quation  曾慶耀教授最佳控制概論上課筆記  Method of Lagrange Multiplier http://www.physics.oregonstate.edu/~tgiebult/COURSES/ ph441/pdf/4410425b.pdf http://www.physics.oregonstate.edu/~tgiebult/COURSES/ ph441/pdf/4410425b.pdf  Euler-Lagrange Equation picture http://borisv.lk.net/matsc597c- 1997/phases/Lecture5/node4.html http://borisv.lk.net/matsc597c- 1997/phases/Lecture5/node4.html  Example https://www.youtube.com/watch?v=H4HN4ZrVm0w https://www.youtube.com/watch?v=H4HN4ZrVm0w 11

12. Thank you for listening~ 12