By Daniel Liberzon
This textbook deals a concise but rigorous creation to calculus of adaptations and optimum regulate idea, and is a self-contained source for graduate scholars in engineering, utilized arithmetic, and similar matters. Designed particularly for a one-semester direction, the e-book starts with calculus of diversifications, getting ready the floor for optimum keep an eye on. It then offers a whole facts of the utmost precept and covers key issues reminiscent of the Hamilton-Jacobi-Bellman conception of dynamic programming and linear-quadratic optimum keep an eye on.
Calculus of adaptations and optimum regulate Theory additionally strains the historic improvement of the topic and contours a number of workouts, notes and references on the finish of every bankruptcy, and recommendations for extra study.
- Offers a concise but rigorous creation
- Requires restricted historical past up to speed idea or complex arithmetic
- Provides a whole evidence of the utmost precept
- Uses constant notation within the exposition of classical and glossy themes
- Traces the ancient improvement of the topic
- Solutions guide (available in simple terms to teachers)
Leading universities that experience followed this publication include:
- University of Illinois at Urbana-Champaign ECE 553: optimal keep watch over platforms
- Georgia Institute of know-how ECE 6553: optimum keep an eye on and Optimization
- college of Pennsylvania ESE 680: optimum keep watch over Theory
- college of Notre Dame EE 60565: optimum Control
Read or Download Calculus of variations and optimal control theory. A concise introduction PDF
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Extra resources for Calculus of variations and optimal control theory. A concise introduction
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.