It should be noted that the quantities y( that we must compute are the values of yt(iR0/n,r) when r = Ri. Since our differential equations have been obtained by letting the upper limits of our integrals be variable so that they might be differentiated, we no longer need this flexibility and from here on, yt will be understood to be the quantity yi(iR0/n,R^. No new notation needs to be set up with this understanding. Equations (14) and (15) represent a system of n +1 simultaneous first-order differential equations so that the problems is now in the standard form for application of optimal control theory. The appropriate Hamiltonian for this system, adjusted for the fact that the final value r = Rx is unknown, is given by In this expression, the X’s are undetermined multipliers which we must calculate. It turns out, as will be seen later, that the nature of this problem is such that kr need not be explicitly computed. In other words, the optimum value of u(r) is a finite sum of zero-order Bessel functions multiplied by various factors. The optimum value of pM is by definition simply the square of the above expression for u. The determination of the X’s will be made subsequently. For the moment, we notice that since all of the unknown X’s (except for the unnecessary Xr) are constants. This is a
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