1. Part 7 Optimization in Functional Space
7.0 Motivation Example

2. Maximizing Yield of Batch Reaction

3. Maximizing Yield of Batch Reaction

4. Part 7 Optimization in Functional Space
7.1 Calculus of Variation

5. Objective Functions

6. Equivalence of Lagrange and Bolza Forms

7. Equivalence of Bolza and Mayer Forms

8. Example

9. Problem Statement

10. Intuitive Interpretation
Let’s visualize a competition, to which only functions which have 2 derivatives in (a,b) and which take on the prescribed end values are permissible. Let’s further assume that there exists a x*(t) such that I is the smallest.

11. Variation

12. Necessary Condition

13. Integration by Parts - 2nd Term

14. Euler-Lagrange Equation

15. Example

16. Transversality Conditions

17. Transversality Conditions

18. Example 1

20. Solution of Example 1

21. Example 2

22. Solution of Example 2

23. Dependent Boundary Conditions

24. Dependent Boundary Conditions

25. Unspecified Terminal Time
We now consider a generalized problem where the final time is defined as the first time after the initial time t0 that the state trajectory is a member of a target set or terminal manifold.

26. Problem Definition

28. Variations of Optimal Trajectory and Terminal Time

29. Necessary Conditions

30. Terminal Constraint

31. Necessary Conditions

32. Example

34. Example

35. Vector Formulation

36. Example

37. 37