[SGVLUG] OT: Hybrid efficiency (was:New Linux Lug)

[David Lawyer]

On Mon, Feb 20, 2006 at 08:06:45PM -0800, Dustin Laurence wrote:
> On Mon, Feb 20, 2006 at 08:00:35PM -0800, Dustin Laurence wrote:
> 
> > No, quite the opposite, and one derivation is a simple but somewhat
> > tedious high-school level exercise.  For simplicity consider driving at
> > each of two velocities u and v for a distance s and compare it with
> > driving at the average speed w for a distance 2s.  We want to take the
> > same time to go the total distance 2s in each case, so we have uv 2s/w =
> > s/u + s/v --> 1/u + 1/v = 2/w or w = 2 ----- v + u
> 
> I don't know what happened there.  It should read
> 
>                                                 uv
> s/u + s/v = 2s/w --> 1/u + 1/v = 2/w <--> w = 2 ---

Here's how I would do this.  Problem, find v(t) so as to minimize the
integral wrt/x of { v^2 } limits: 0, L is length of trip, where T is
the fixed time for the trip so that x(T) = L

Solution: Use optimal control theory.  Make the independent var time.
Since dx = v*dt the problem becomes to minimize Integral wrt/t { v^2 * v }
= { v^3 } limits: 0, T.  By definition of velocity 
dx/dt = v Now the variational Hamiltonian is 
H= v^3 + l(t)v(t)
where l() is the multiplier function.  Minimizing the Hamiltonian wrt
v, the control variable, by setting the partial derivative = 0  
Hv = 3*v^2 + l = 0  (Hv is the partial derivative of H wrt v)
Now the adjoint equation is:
dl/dt = -Hx (where Hx is the partial derivative of H wrt x) = 0 since
H doesn't depend on x.  So this implies l(t) must be constant for all
t, which implies that v(t) must be constant for all t since Hv = 0 (see
above).  QED.
			David Lawyer