Charged particle beam optimization with use of a second order method

The problem of optimization of charged particle beam dynamics is considered. The problem is formulated as the control problem for a dynamical system ensemble with a fixed endpoint. A state of the dynamical system ensemble is described by a density of the systems in the phase space, which satisfies to the Liouville equation or to the Vlasov equation. The problem is to minimize a functional depending on terminal state of the ensemble.
It is proposed to use an algorithm based on calculation of the first and the second variations of trajectory of a dynamical system under the control function variation. If the control function is parametrized, expressions for the first and the second variations allow to find the first and the second derivatives of the functional being minimized over control parameters. Using of the second derivatives can make the optimization process sufficiently quicker, as compared with algorithm using only first derivatives.
The proposed algoritm is realized for a beam in the Radio Frequency Quadrupole (RFQ) channel, which is often used as initial part of a charged particles accelerator. The simplest problem of of optimization of longitudinal dynamics of the beam in this channel is considered. The numerical solution is finding on the base of the method of macroparticles. The comparison between the first order and the second order methods is conducted. The second order method shows sufficient increase of the rate of convergence as compared with the first order method.