Abstract:
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.
Keywords:optimal control, dynamical system ensemble, second order variation, second order method, charged particle beam.