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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 8 Properties of Optimal Control Solution Bryson and Ho – Section 3.5 and Kirk – Section 4.4 Spr 2008 16.323 8– Properties of Optimal Control • If g = g ( x , u ) and a = a ( x , u ) do not explicitly depend on time t , then the Hamiltonian H is at least piecewise constant. H = g ( x , u ) + p T a ( x , u ) (8.1) then dH ∂H ∂H d x ∂H d u ∂H d p = + + + (8.2) dt ∂t ∂ x dt ∂ u dt ∂ p dt = H x a + H u u ˙ + H p p ˙ (8.3) Now use the necessary conditions: x ˙ = a = H p T (8.4) p ˙ = − H T (8.5) x to get that dH = − p ˙ T a + a T p ˙ + H u u ˙ = H u u ˙ dt Third necessary condition requires H u = 0 , so clearly dH = 0 , which • dt suggests H is a constant, – Note that it might be possible for the value of this constant to change at a discontinuity of u , since then u ˙ would be infinite, and · ∞ is not defined....
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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.

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lec8 - MIT OpenCourseWare http/ocw.mit.edu 16.323...

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