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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 .
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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 0 x ( t ) αδ x (1) -αδ x (1) Figure by MIT OpenCourseWare.
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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 0 subject to x ˙ = f ( x , u , t ) x ( t 0 ) , t 0 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
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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 ) � ≥ 0 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 0 June 18, 2008
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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. An increment of a functional Δ J ( x ( t ) , δ x ( t )) = J ( x ( t ) + δ x ( t )) J ( x ( t )) A variation of the functional is a linear approximation of this increment: Δ J ( x ( t ) , δ x ( t )) = δJ ( x ( t ) , δ x ( t )) + H.O.T.
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