By John Gregory Ph.D., Cantian Lin Ph.D. (auth.)
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Additional resources for Constrained Optimization in the Calculus of Variations and Optimal Control Theory
3 above. 2). 4. As an example of a very general theory, we show that if b > 11', then (-l, - -l) J(y) = fob (y'2 - y2)dx such that y(O) = y(b) = 0 has no minimum solution. 6 using a different method. To see this result, we note that y == 0 implies J(y) = O. If the minimum solution is J(yo) < 0, then by the quadratic nature of J, J(2yo) = 4J(yo) < J(yo). Thus, if Yo is a minimum solution, we must have J(yo) = o. Also, if Yo is a minimum solution, it is a critical point solution and hence the 42 Chapter 2.
This problem has a long and important history (see Gregory ). We will see that the Euler-Lagrange equation for this problem is a linear, selfadjoint, second order ordinary differential equation, and hence relatively easy to handle. In addition, parameterized problems of this type lead to the standard eigenvalue problems of ordinary differential equations. 6). 12) with A = B = 0 which is often called the second (or accessory) variational problem. 1) and since I" (yO, 0) = 0, we must show that the minimulll value of I" (Yo, z) is zero for all z such that z(a) = z(b) = O.
The major result is that a critical point solution yo(x) satisfies the Euler-Lagrange equation, which is a second order ordinary differential equation. This result is the essential tool used to construct analytic solutions when possible. 1. This will allow the reader to become acquainted with some historical ideas as well as to formulate and solve basic problems. 1). We obtain four types of Euler-Lagrange equations depending on the smoothness of our problem, as well as corner conditions and transversality conditions which are necessary conditions for a critical 23 24 Chapter 2.
Constrained Optimization in the Calculus of Variations and Optimal Control Theory by John Gregory Ph.D., Cantian Lin Ph.D. (auth.)