February 25, 2017

Constrained Optimization in the Calculus of Variations and by John Gregory Ph.D., Cantian Lin Ph.D. (auth.)

By John Gregory Ph.D., Cantian Lin Ph.D. (auth.)

Show description

Read Online or Download Constrained Optimization in the Calculus of Variations and Optimal Control Theory PDF

Similar calculus books

Tensors, differential forms, and variational principles

Incisive, self-contained account of tensor research and the calculus of external differential types, interplay among the idea that of invariance and the calculus of diversifications. Emphasis is on analytical suggestions, with huge variety of difficulties, from regimen manipulative workouts to technically tricky assignments.

Q-valued functions revisited

Quantity 211, quantity 991 (first of five numbers).

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

Sample text

30b). 29b). (f) + Ja f(x, y + fz, y' + Ez')dx. Using Leibnitz's rule as before we have 0= F' (0) = I' (y, z) = gx(b, y(b)) . 29b) leads to + gy(b,y(b))y'(b) + f(b,Y(b)'Y'(b))l¢'(b;~)Y'(b) + gy (b, y(b ))z(b) + z(b)fy, (b, y(b), y' (b)) = 0 [gx(b,y(b)) or gx + gyY' + f + [gy + fy, H¢' - y'li (b,y(b),y'(b)) = o. 31). 32) f + fy, [¢' - y'll (b,y(b),y'(b)) = O. 33) gx + gy¢' + f + fy, [¢' - y'll (b,y(b),y'(b)) = O. Thus, we have the definition of critical point solutions, below. 48 Chapter 2. 32). 33).

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}. 21) and all subsequent results must hold. 26). 2 and leave all subsequent results (including the corner conditions) to the reader. 21) except for at most a finite number of interior points.

3 with J 2(x, y) = yT Ax where A is a symmetric n x n matrix. Show that J2(x) = h(x, x) has a strict global minimum at x = 0 if A is positive definite. 32 Chapter 2. 4 is negative then J2(X) has no local minimum by considering the Taylor series expansion for J(x + EY) where y =f 0 is an eigenvector corresponding to a negative eigenvalue. Some important ideas of quadratic forms are best illustrated in this context by two simple examples. 12). In this case, J 1 (y} = J 1 (y,z) = J 1 (y + EZ) 1 b (y'2+ y2)dx, ~J~(Y,z) = Io b (ylZI +yz)dx and = J1 (y) + 2EJ1 (y, z) + E2 J1 (z).

Download PDF sample

Rated 4.90 of 5 – based on 6 votes