Lectur19 - Lecture XIX Euler Equations and a Basic Example...

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Lecture XIX Euler Equations and a Basic Example I. Review A. Concepts borrowed from calculus 1. Maximum/Minimum a. The variable x * maximizes f(x) if f(x * )-f(x) > 0 for all x in the neighborhood of x * . b. The function y 0 (x) maximizes I[y] if I[y 0 ]-I[y] > 0 for all functions in the neighborhood of y 0 . 2. Continuity a. Univariate: lim ( ) lim ( ) ( ) xx fx fx fx →→ -+ == 00 0 b. Continuity of functional I[y]: If ε >0 there is a δ >0 such that |y 0 (x)-y(x)| < δ implies that |I[y 0 (x)]-I[y(x)]|< ε , then I[y] is continuous at y 0 (x). c. Why are we concerned? If the functional is continuous around y 0 (x), the optimal path, then we can use concepts like differential calculus to determine the optimal path. 3. Norms a. The norm of a function is the largest absolute value that a function obtains over a stated range. b. The difference between two functions is then defined as the largest absolute difference between the two functions over a stated range. c. Extending the concept of the norm ||y(x)|| 0 is the largest absolute value that the function y(x) obtains ||y(x)|| 1 is the largest absolute value of the function plus the largest absolute value of the first derivative. B. Derivation of the Necessary Conditions 1. In the Lecture XVIII we developed the general calculus of variation problem as: max [ ] ( , ( ), ’( )) () , Iy f xyx y x d x yx y yx y x x = 0 1 11 We assumed that there existed an optimal path y(x). Further, we assumed z(x) was in the neighborhood of y(x). z(x) was constructed as a deviation of y(x): zx =+ =- δ δ Note that if z(x) is near to y(x), z(x) approaches y(x), and z’(x) approaches y’(x). Thus, both δ y(x) and ( δ y(x)) become very small.
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2. Again changing notation δε φ yx x () = where φ (x 0 )= φ (x 1 )=0. Further, ε is a scalar that we can let approach zero. Thus, the alternative feasible path can be expressed as zx x =+ ε φ The functional value as a function of the alternative feasible path can then be expressed as: If x y x x y x x d x x x o + (,() ,’ ) εφ εφ 1 3. To derive the maximum, we will differentiate the functional expressed with respect to the alternative path with respect to ε . To be technically correct, we must first define two new functions Φ(ε29 and Φ (ε29 based on the z(x) and z’(x) functions. Specifically, x x ’( ) ⇒= + + Φ Φ εε φ φ The above functional then becomes x d x x x = (, () , ’ ) ΦΦ 0 1
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Lectur19 - Lecture XIX Euler Equations and a Basic Example...

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