## John Gregory Ph.D., Cantian Lin Ph.D. (auth.)'s Constrained Optimization in the Calculus of Variations and PDF By John Gregory Ph.D., Cantian Lin Ph.D. (auth.)

ISBN-10: 9401052956

ISBN-13: 9789401052955

ISBN-10: 9401129185

ISBN-13: 9789401129183

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Extra resources for Constrained Optimization in the Calculus of Variations and Optimal Control Theory

Example text

For convenience, we assume our earlier notation with the two points (a, 0) and (b, B) where a > band B > O. 2 above. To obtain a mathematical formulation we must first consider some physical ideas. Let s denote the distance traveled, t denote time and v = v(t), the velocity. Then we have from the principle of the conservation of energy that 1 ds "2mv2 = mgy where v = dt. 9) minimize t = l b 29 VI + y'2(x) a such that y(a) = 0, V dx 2gy(x) y(b) = B. ~ Once again we note that while there are many necessary and sufficient conditions for the general problem in the calculus of variations, physical intuition suggests that the solution of an Euler-Lagrange equation is the unique solution to this problem.

24) follow as above. We now turn our attention to the corner conditions. Since we will not need these conditions for our numerical work later in this book, we first consider why they are even necessary for our discussion of classical results. The key idea is the order of the Euler-Lagrange equation. Perhaps a simple example will best explain what is happening. We know that y' = y is a first order differential equation with a one parameter family solution space y(x) = ce X. Thus, for example, if we require y(O) = 1, the unique solution will be yo(x) = eX.

This is what the transversality conditions do. In practice, we will use transversality conditions in two situations. The first situation is when a and b are given but y(a) and/or y(b) are not. 4 above. To avoid being tedious, we assume that (a, y( a)) is given in both these situations and leave the remaining cases to the reader. 26) 1 43 b minimize I(y) = such that y( a) = A, f(x, y, y')dx b is given. 4, which motivates our results, is shown above. 21) above, except that we consider a fixed variation z E Za = {y E Y : y(a) = O}.