By Daniel Liberzon
This textbook deals a concise but rigorous advent to calculus of adaptations and optimum keep an eye on concept, and is a self-contained source for graduate scholars in engineering, utilized arithmetic, and comparable matters. Designed particularly for a one-semester path, the ebook starts with calculus of adaptations, getting ready the floor for optimum keep an eye on. It then offers a whole evidence of the utmost precept and covers key issues similar to the Hamilton-Jacobi-Bellman conception of dynamic programming and linear-quadratic optimum keep an eye on.
Calculus of adaptations and optimum regulate Theory additionally lines the old improvement of the topic and contours a variety of workouts, notes and references on the finish of every bankruptcy, and recommendations for additional study.
- Offers a concise but rigorous creation
- Requires restricted history on top of things conception or complex arithmetic
- Provides an entire 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 merely to teachers)
Leading universities that experience followed this e-book include:
- University of Illinois at Urbana-Champaign ECE 553: optimal keep an eye on structures
- Georgia Institute of know-how ECE 6553: optimum keep watch over and Optimization
- collage 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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Additional info for Calculus of variations and optimal control theory. A concise introduction
Example text
This b gives a ξ(x)η(x)dx > 0, and we reach a contradiction. 1 that for y(·) to be an extremum, a necessary condition is Ly (x, y(x), y (x)) = d Lz (x, y(x), y (x)) dx ∀ x ∈ [a, b]. 17) This is the celebrated Euler-Lagrange equation providing the first-order necessary condition for optimality. 17): y and y are treated as independent variables when computing the partial derivatives L y and Ly , then one plugs in for these variables the position y(x) and velocity y (x) of the curve, and finally the differentiation with respect to x is performed using the chain rule.
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. 18) was derived by Euler around 1740. His original derivation was very different from the one we gave, and relied on discretization.
7). For example, we can set η(x) = (x − c) 2 (x − d)2 for x ∈ [c, d] and η(x) = 0 otherwise. This b gives a ξ(x)η(x)dx > 0, and we reach a contradiction. 1 that for y(·) to be an extremum, a necessary condition is Ly (x, y(x), y (x)) = d Lz (x, y(x), y (x)) dx ∀ x ∈ [a, b]. 17) This is the celebrated Euler-Lagrange equation providing the first-order necessary condition for optimality. 17): y and y are treated as independent variables when computing the partial derivatives L y and Ly , then one plugs in for these variables the position y(x) and velocity y (x) of the curve, and finally the differentiation with respect to x is performed using the chain rule.