# Calculus of Variations and Optimal Control Theory: A Concise by Daniel Liberzon

By Daniel Liberzon

This textbook bargains a concise but rigorous creation to calculus of diversifications and optimum keep an eye on thought, and is a self-contained source for graduate scholars in engineering, utilized arithmetic, and similar topics. Designed particularly for a one-semester path, the ebook starts off with calculus of diversifications, getting ready the floor for optimum keep watch over. It then supplies an entire evidence of the utmost precept and covers key subject matters reminiscent of the Hamilton-Jacobi-Bellman concept of dynamic programming and linear-quadratic optimum keep an eye on.

Calculus of adaptations and optimum regulate Theory additionally lines the ancient improvement of the topic and contours various workouts, notes and references on the finish of every bankruptcy, and recommendations for extra study.

• Offers a concise but rigorous advent
• Requires restricted heritage up to speed concept or complex arithmetic
• Provides an entire facts of the utmost precept
• Uses constant notation within the exposition of classical and smooth subject matters
• Traces the ancient improvement of the topic
• Solutions handbook (available merely to teachers)

Leading universities that experience followed this publication include:

• University of Illinois at Urbana-Champaign ECE 553: optimal keep watch over structures
• Georgia Institute of expertise ECE 6553: optimum keep watch over and Optimization
• collage of Pennsylvania ESE 680: optimum regulate Theory
• college of Notre Dame EE 60565: optimum Control

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Additional info for Calculus of Variations and Optimal Control Theory: A Concise Introduction

Sample text

Of course the answer is obvious, but let us see how we can obtain it from the Euler-Lagrange equation. 1, in the context of Dido’s problem and the catenary problem (where it played the role of a side constraint rather than b the cost functional). This length functional is J(y) = a 1 + (y (x))2 dx, √ hence the Lagrangian is L(x, y, z) = 1 + z 2 . Since Ly = 0, the Eulerd Lz (x, y(x), y (x)) = 0. This implies that Lz stays Lagrange equation reads dx √ constant along the shortest path. We have Lz = z/ 1 + z 2 , thus Lz (x, y(x), y (x)) = y (x) 1 + (y (x))2 .

We can allow y to be discontinuous at some points; we will discuss such situations and see their importance soon. The function L is called the Lagrangian, or the running cost. It is clear that a maximization problem can always be converted into a minimization problem by flipping the sign of L. In the analysis that follows, it will be important to remember the following point: Even though y and y are the position and velocity along the curve, L is to be viewed as a function of three independent variables.

24) cvoc-formatted August 24, 2011 14 7x10 CHAPTER 1 This is the first-order necessary condition for constrained optimality. The coefficients λ∗i , i = 1, . . , m are called Lagrange multipliers. 2 Give an example where a local minimum x∗ is not a regular point and the above necessary condition is false (be sure to justify both of these claims). The above proof of the first-order necessary condition for constrained optimality involves geometric concepts. 20). We now give a shorter alternative proof which is purely analytic, and which will be useful when we study problems with constraints in calculus of variations.