By John H. Mathews, Russell W. Howell
This article is meant for undergraduate, scholars in arithmetic, physics, and engineering. it truly is meant to stability among the natural and utilized elements of complicated research and to offer ideas in a transparent writing type that's comprehensible to scholars on the junior or senior undergraduate point. A wealth of workouts that adjust in either hassle and substance offers the textual content flexibility. adequate purposes are incorporated to demonstrate how complicated research is utilized in technological know-how and engineering.
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Example text
In general, the question of existence of optimal solutions is not trivial, and the following example should serve as a warning. 3 Consider the problem of minimizing J(y) = 0 y(x)(y (x))2 dx subject to the boundd ary conditions y(0) = y(1) = 0. The Euler-Lagrange equation is dx (2yy ) = (y )2 , and y ≡ 0 is a solution. Actually, one can show that this is a unique extremal satisfying the boundary conditions (we leave the proof of this fact to the reader). But y ≡ 0 is easily seen to be neither a minimum nor a maximum.
Similar remarks apply to other necessary conditions to be derived in this chapter. 2, on the other hand, will apply to weak minima only. 1 Euler-Lagrange equation We continue to follow the notational convention of Chapter 1 and denote by Lx , Ly , Lz , Lxx , Lxy , etc. the partial derivatives of the Lagrangian L = L(x, y, z). To keep things simple, we assume that all derivatives appearing in our calculations exist and are continuous. 3. Let y = y(x) be a given test curve in A. 8). Clearly, this is true if and only if η(a) = η(b) = 0.
Dx a This conclusion can be summarized as follows: b a Ly − d Ly dx b η(x)dx = 0 ∀ η such that a My − d My dx η(x)dx = 0. 21) on page 10. 49) CALCULUS OF VARIATIONS AND OPTIMAL CONTROL 43 for all x ∈ [a, b]. Rearranging terms, we see that this is equivalent to d (L + λ∗ M )y dx which amounts to saying that the Euler-Lagrange equation holds for the augmented Lagrangian L + λ∗ M . In other words, y is an extremal of the augmented cost functional (L + λ∗ M )y = b (J + λ∗ C)(y) = L(x, y(x), y (x)) + λ∗ M (x, y(x), y (x)) dx.