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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 16.323 Principles of Optimal Control Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 16.323 Lecture 5 Calculus of Variations Calculus of Variations • • Most books cover this material well, but Kirk Chapter 4 does a particularly nice job. See here for online reference. • x * x *+ αδ x (1) x * αδ x (1) t t f t x ( t ) αδ x (1)αδ x (1) Figure by MIT OpenCourseWare. Spr 2008 16.323 5–1 Calculus of Variations • Goal: Develop alternative approach to solve general optimization problems for continuous systems – variational calculus – Formal approach will provide new insights for constrained solutions, and a more direct path to the solution for other problems. • Main issue – General control problem, the cost is a function of functions x ( t ) and u ( t ) . t f min J = h ( x ( t f )) + g ( x ( t ) , u ( t ) ,t )) dt t subject to x ˙ = f ( x , u ,t ) x ( t ) ,t given m ( x ( t f ) ,t f ) = 0 – Call J ( x ( t ) , u ( t )) a functional . • Need to investigate how to find the optimal values of a functional. – For a function, we found the gradient, and set it to zero to find the stationary points, and then investigated the higher order derivatives to determine if it is a maximum or minimum. – Will investigate something similar for functionals. June 18, 2008 Spr 2008 16.323 5–2 Maximum and Minimum of a Function • – A function f ( x ) has a local minimum at x if f ( x ) ≥ f ( x ) for all admissible x in x − x ≤ – Minimum can occur at (i) stationary point, (ii) at a boundary, or (iii) a point of discontinuous derivative. – If only consider stationary points of the differentiable function f ( x ) , then statement equivalent to requiring that differential of f satisfy: ∂f df = d x = 0 ∂ x for all small d x , which gives the same necessary condition from Lecture 1 ∂f = 0 ∂ x • Note that this definition used norms to compare two vectors. Can do the same thing with functions distance between two functions ⇒ d = x 2 ( t ) − x 1 ( t ) where 1. x ( t ) ≥ for all x ( t ) , and x ( t ) = 0 only if x ( t ) = 0 for all t in the interval of definition. 2. a x ( t ) =  a  x ( t ) for all real scalars a . 3. x 1 ( t ) + x 2 ( t ) ≤ x 1 ( t ) + x 2 ( t ) Common function norm: • t f 1 / 2 x ( t ) 2 = x ( t ) T x ( t ) dt t June 18, 2008 Spr 2008 16.323 5–3 Maximum and Minimum of a Functional • – A functional J ( x ( t )) has a local minimum at x ( t ) if J ( x ( t )) ≥ J ( x ( t )) for all admissible x ( t ) in x ( t ) − x ( t ) ≤ • Now define something equivalent to the differential of a function called a variation of a functional....
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This note was uploaded on 11/07/2011 for the course AERO 16.323 taught by Professor Jonathanhow during the Spring '08 term at MIT.
 Spring '08
 jonathanhow

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