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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 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 0 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 0 · ∞ is not defined. Thus H is at least piecewise constant For free final time problems, transversality condition gives, h t + H ( t f ) = 0 .

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