Charged particles dynamics optimization in a drift-tube linear accelerator
A problem of charged particle dynamics optimization in a drift-tube linear accelerator is considered. Discrete optimization model based on the Solovjev equations is suggested. Functional is considered for dynamics estimation to allow conducting optimization and taking into account a density distri...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2006
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nasplib_isofts_kiev_ua-123456789-788792025-02-09T13:41:57Z Charged particles dynamics optimization in a drift-tube linear accelerator Оптимизация динамики заряженных частиц в ускорителе с трубками дрейфа Оптимізація динаміки заряджених часток у прискорювачі з трубками дрейфу Kotina, E.D. Линейные ускорители заряженных частиц A problem of charged particle dynamics optimization in a drift-tube linear accelerator is considered. Discrete optimization model based on the Solovjev equations is suggested. Functional is considered for dynamics estimation to allow conducting optimization and taking into account a density distribution of charged particles. Analytical expression for functional variation that helps constructing of various directed methods of optimization is suggested. Рассматривается проблема оптимизации динамики заряженных частиц в линейном ускорителе с трубками дрейфа. Предлагается дискретная модель оптимизации, основанная на уравнениях Соловьева. Для оценки динамики вводится функционал, позволяющий проводить совместную оптимизацию программного и возмущенных движений с учетом плотности распределения частиц в фазовом пространстве. Выписывается аналитическое представление вариации предложенного функционала, дающее возможность построения направленных методов оптимизации. Розглядається проблема оптимізації динаміки заряджених часток у лінійному прискорювачі з трубками дрейфу. Пропонується дискретна модель оптимізації, заснована на рівняннях Соловйова. Для оцінки динаміки вводиться функціонал, що дозволяє проводити спільну оптимізацію програмного і збуреного рухів з урахуванням густини розподілу часток у фазовому просторі. Виписується аналітичне зображення варіації запропонованого функціонала, що дає можливість побудови спрямованих методів оптимізації. The Russian Fond of Fundamental Researches, project 03-01-00726, supported this work. 2006 Article Charged particles dynamics optimization in a drift-tube linear accelerator / E.D. Kotina // Вопросы атомной науки и техники. — 2006. — № 2. — С. 137-139. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 517.97:621.384.6 https://nasplib.isofts.kiev.ua/handle/123456789/78879 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Линейные ускорители заряженных частиц Линейные ускорители заряженных частиц |
| spellingShingle |
Линейные ускорители заряженных частиц Линейные ускорители заряженных частиц Kotina, E.D. Charged particles dynamics optimization in a drift-tube linear accelerator Вопросы атомной науки и техники |
| description |
A problem of charged particle dynamics optimization in a drift-tube linear accelerator is considered. Discrete optimization
model based on the Solovjev equations is suggested. Functional is considered for dynamics estimation to
allow conducting optimization and taking into account a density distribution of charged particles. Analytical expression
for functional variation that helps constructing of various directed methods of optimization is suggested. |
| format |
Article |
| author |
Kotina, E.D. |
| author_facet |
Kotina, E.D. |
| author_sort |
Kotina, E.D. |
| title |
Charged particles dynamics optimization in a drift-tube linear accelerator |
| title_short |
Charged particles dynamics optimization in a drift-tube linear accelerator |
| title_full |
Charged particles dynamics optimization in a drift-tube linear accelerator |
| title_fullStr |
Charged particles dynamics optimization in a drift-tube linear accelerator |
| title_full_unstemmed |
Charged particles dynamics optimization in a drift-tube linear accelerator |
| title_sort |
charged particles dynamics optimization in a drift-tube linear accelerator |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2006 |
| topic_facet |
Линейные ускорители заряженных частиц |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/78879 |
| citation_txt |
Charged particles dynamics optimization in a drift-tube linear accelerator / E.D. Kotina // Вопросы атомной науки и техники. — 2006. — № 2. — С. 137-139. — Бібліогр.: 5 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
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| first_indexed |
2025-11-26T09:33:05Z |
| last_indexed |
2025-11-26T09:33:05Z |
| _version_ |
1849844927440093184 |
| fulltext |
CHARGED PARTICLES DYNAMICS OPTIMIZATION
IN A DRIFT-TUBE LINEAR ACCELERATOR
E.D. Kotina
Saint-Petersburg State University, Saint-Petersburg, RUSSIA
A problem of charged particle dynamics optimization in a drift-tube linear accelerator is considered. Discrete op-
timization model based on the Solovjev equations is suggested. Functional is considered for dynamics estimation to
allow conducting optimization and taking into account a density distribution of charged particles. Analytical expres-
sion for functional variation that helps constructing of various directed methods of optimization is suggested.
PACS: 517.97:621.384.6
1. INTRODUCTION
We consider this problem as a non-standard problem
of the theory of optimal control in discrete systems. This
problem of control of particular trajectory and ensem-
bles of trajectories (beams of trajectories) had been con-
sidered under various criteria of quality. Along trajecto-
ries of the system we consider functional characterizing
the dynamics of programmed motion (motion of syn-
chronous particle) as well as dynamics of perturbed mo-
tion. The mathematical model suggested in this work
also allows accounting for a density of particle distribu-
tion in phase space. The method of solving of this prob-
lem is proposed.
2. MATHEMATICAL MODEL
We consider the problem of optimizing of the beam
parameters for the drift-tube linear accelerator. To char-
acterize the motion in this accelerator we have applied
the discrete Solovjev equations [1].
In common case, Solovjev equations can be written
in the following form:
)),(),(,()1( kukxkfkx =+ (1)
)),(),(),(,()1( kukykxkFky =+ (2)
for 1,...,0 −= Nk , where )(kx is the −n dimensional
phase vector defining programmed motion, )(ky is the
−m dimensional phase vector defining perturbed mo-
tion, )(ku is −r dimensional control vector, and
))(),(,( kukxkf is the −n dimensional vector function
defining the process dynamics at each step. For all
{ }Nk ,,1,0 ∈ the vector function ))(),(,( kukxkf is
assumed to be definite and continuous on )(kUx ×Ω at
all its arguments ))(),(( kukx , along with partial deriva-
tives with respect to these variables.
))(),(),(,( kukykxkF is the −m dimensional vector
function; for all { }Nk ,,1,0 ∈ it is assumed to be defi-
nite and continuous on )(kUyx ×Ω×Ω at all its argu-
ments ))(),(),(( kukykx , along with the partial deriva-
tives with respect to these variables and second partial
derivatives. xΩ – is the domain in nR , yΩ is the do-
main in mR , and 1,,1,0),( −= NkkU is a compact
set in .rR In this case we consider the Jacobian func-
tion
)(
)())(),(),(,(
ky
kFkukykxkJJ k ∂
∂==
to be non-zero for all changes of )(),(, kykxk and )(ku .
Eq. (1) describes the dynamics of programmed motion.
Eq. (2) describes the dynamics of perturbed motions.
We assume that 0)0( xx = is fixed and the initial
state of system (2) is described by set −0M a compact
set of non-zero measures in mR . We call the sequence
of vectors { })1(,),1(),0( −Nuuu the control of the
system described by Eqs. (1) and (2) and denote it by u
for brevity. We call the corresponding sequence of vec-
tors { })(,),1(),0( Nxxx the trajectory of programmed
motion and denote it by ),( 0 uxxx = . We denote the
phase state of a programmed trajectory for the −k th
step by ))(,,()( 0 kuxkxkx = . Similarly, we call the se-
quence of vectors { })(,),1(),0( Nyyy the trajectory of
perturbed motion and denote it by ),,( 0 uyxyy = . We
denote the phase state of the particle for the −k th step
by ),,,()( 0 uyxkyky = . The set of trajectories
),,( 0 uyxy corresponding to the initial state 0x , the
control u and different states 00 My ∈ are referred to
as an ensemble of trajectories, or the beam of trajecto-
ries. The phase state of the beam at the −k th step is
also called the cross-section of the beam of trajectories
and is denoted by ukM , i.e.:
{ },)),(),(,,()(:)( 000, MykukxykykykyM uk ∈==
and the controls satisfying conditions
1...,1,0),()( −=∈ NkkUku are admissible.
Let 0x be the initial state of a synchronous particle
and 0M be the set of initial phase states of charged par-
ticles with density distribution ),0()( 000 yy ρρ = . We
would like to determine how the distribution density
function along the beam trajectories is transformed. Let
us fix an instant 1+k and a point ukk My ,11 ++ ∈ . Let
)1( +ky be an image of the point )(ky in view of
Eq. (2). We denote by ))(( kyG the set of points
uk
i Mky ,)( ∈ , such that the trajectories of the system
emanating at the instant k from )(kyi at the step 1+k
__________________________________________________________
PROBLEMS OF ATOMIC SСIENCE AND TECHNOLOGY. 2006. № 2.
Series: Nuclear Physics Investigations (46), p.137-139. 137
fall within a certain −r neighborhood ))1(( +kyS r of
the point )1( +ky .
By the distribution density of trajectories from
Eq. (2) at the point )1( +ky on the 1+k - th step, we
mean the limit:
( )( ) ( )
( )
1 0
( )
1( 1, ) lim , d ,
( 1)k k kr
r G y k
k y k y y
mes S y k
ρ ρ+ →
м ьп п+ = н э+п по ю
т (3)
where
( ) .)1())1((
))1((
∫
+
+=+
kyS
r
r
kdykySmes
From the one-to-one correspondence of the sets
))(( kyG and ))1(( +kyS r , the integral in Eq. (3) trans-
forms to the form:
)1(),(
))1((
1 +∫
+
− kdyJyk
kyS
kk
r
ρ , (4)
where )(kyyk = and
( )
( ) .
)(
)(),(),(,det
)1(
)(det
)(),(),(,
1
11
∂
∂
=
+∂
∂
==
−
−−
ky
kukykxkF
ky
ky
kukykxkJJ k
In view of this and the form ( )))1(( +kySmes r , we ob-
tain the following equation for ),( kykρ :
).(),0(),,())1(,1( 000
1 yyykJkyk kk ρρρρ ==++ − (5)
The function ),()( kykk ρρ = denotes the distribution
density function for the k -th step.
We introduce the following functional:
,),((
)),,(,,()(
,
,
,
1
1
∫
∑ ∫ +=
−
=
uN
uk
M
NNN
N
k M
kkkkkk
dyyNyg
dyuykyxuI
ρ
ρϕ
(6)
where kϕ and g are continuously differentiable func-
tions, )(kxxk = .
The functional (6) allows the simultaneous estima-
tion of programmed and perturbed motions, as well as
their simultaneous optimization and taking into account
the density of particle distribution in the phase space.
3. VARIATION OF FUNCTIONAL
Let us rewrite Eqs. (1) and (2) and an equation for
the distribution density along trajectories of the system
(2):
)),(),(,()1( kukxkfkx =+
)),(),(),(,()1( kukykxkFky =+ (7)
),())(),(),(,()1( 1 kkukykxkJk ρρ ⋅=+ −
for .1,,0 −= Nk
Let us denote variations of trajectories of system (7)
as )(),( kykx δδ , and )(kδ ρ , with admissible variation of
control u∆ and a given u .
Now we define the corresponding equations for vari-
ations:
),(
)(
)(
)(
)(
)(
)1( ku
ku
kfkx
kx
kfkx ∆
∂
∂
+
∂
∂
=+ δδ (8)
)(
)(
)()(
)(
)()(
)(
)()1( ku
ku
kFky
ky
kFkx
kx
kFky ∆
∂
∂+
∂
∂+
∂
∂=+ δδδ , (9)
).(
)(
)()()()(
)(
)(
)()()(
)(
)()()1(
1
1
11
ku
ku
kJkkkJ
ky
ky
kJkkx
kx
kJkk
∆
∂
∂
+
+
∂
∂
+
∂
∂
=+
−
−
−−
ρδ ρ
δρδρδ ρ
(10)
We also have the following Eqs. [2-3]:
,)(
)(
)()(
)(
)()(
)(
)()(
)()1(
1
∆
∂
∂+
∂
∂+
∂
∂
+=+
− ku
ku
kJkx
kx
kJky
ky
kJkJ
kydivkydiv yy
δδ
δδ
(11)
where
∑
= ∂
∂=
m
i i
i
y ky
kykydiv
1
.
)(
)()( δδ
Taking into account Eqs. (8)–(11), the initial values
of variations 0)0(,0)0(,0)0( === δ ρδδ yx ,
,0)0( =ydivyδ and using methods of investigation for
functions of type (6) [2-3], variation of functional (6)
(for admissible variation of control ∆u) can be repre-
sented in the following form:
),(
)(
)(
)(
)(
)1(
)(
)(
)()1(
)(
)(
)1()(
)(
)(
)1()(
1
1
0 ,
kudy
ku
k
ku
kJkq
ku
kJkkJ
ku
kfkkJ
ku
kFkpkJI
k
k
T
N
k M
T
uk
∆
∂
∂
+
∂
∂
+
+
∂
∂
++
∂
∂
+
+
∂
∂
+=
−
−
=
∑ ∫
ϕ
ρξγ
δ
(12)
where )(),(),( kkkp ξγ and )(kq are the following aux-
iliary functions:
,
)(
),(
)(
∂
∂
=
Ny
ygNp NNT ρ
,
)(
)),(
)(
∂
∂
=
N
yg
N NN
ρ
ρ
ξ
,0)(),,()( == NygNq NN γρ
)()1()()( kkqkJkq ϕ++= , ,
)(
)(
)1()(
k
k
kk
ρ
ϕ
ξξ
∂
∂
++=
,
)(
)(
)(
)(
)1(
)(
)(
)()1()(
)(
)(
)1()()(
1
ky
k
ky
kJ
kq
ky
kJ
kkkJ
ky
kF
kpkJkp TT
∂
∂
+
∂
∂
++
∂
∂
+
+
∂
∂
+=
− ϕ
ρξ
,
)(
)(
)(
)(
)1(
)(
)(
)()1()(
)(
)(
)1()(
)(
)(
)1()()(
1
kx
k
kx
kJ
kq
kx
kJ
kkkJ
kx
kf
kkJ
kx
kF
kpkJk TTT
∂
∂
+
∂
∂
++
∂
∂
+
+
∂
∂
++
∂
∂
+=
− ϕ
ρξ
γγ
for .1,,1 −= Nk
__________________________________________________________
PROBLEMS OF ATOMIC SСIENCE AND TECHNOLOGY. 2006. № 2.
Series: Nuclear Physics Investigations (46), p.137-139. 137
Eq. (12) for functional variation allows the construc-
tion of various methods of optimization of the function-
al in Eq. (6).
4. CONCLUSION
Simultaneous optimization of programmed and per-
turbed motions under various quality criteria was con-
sidered in previous works [3–5] and the results were ap-
plied to the optimization of beam dynamics in linear-
tube accelerators [4-5]. The analytical representation
obtained in this paper for variation of the functional ex-
amined allows taking into account the density distribu-
tion of charged particles as well.
5. ACKNOWLEDGEMENTS
The Russian Fond of Fundamental Researches,
project 03-01-00726, supported this work.
REFERENCES
1. B.P. Murin, B.I. Bondarev, V.V. Kushin, A.P. Fe-
dotov. Ion linear accelerators. V.1. Problems and
theory. - М.: “Atomizdat”, 1978, p.264.
2. D.A. Ovsyannikov, Modeling and Optimization of
Charged Particle Beam Dynamics. Leningrad State
University Publishing House, Leningrad, 1990 (in
Russian).
3. E.D. Kotina, A.D. Ovsyannikov. On simultaneous
optimization of programmed and perturbed mo-
tions in discrete systems. Proc. of the 11th Interna-
tional IFAC Workshop. 2001, v.1, Oxford, UK, p.
187-189.
4. E.D. Kotina, S.A Garbuzova. Optimization of Lon-
gitudinal Motion of Charged Particles in Drift-
Tube Linear Accelerator. Proc. International Work-
shop: Beam Dynamics & Optimization, BDO-2002.
2002, St. Petersburg, p.135-141.
5. E.D. Kotina. Control discrete systems and their ap-
plications to beam dynamics optimization. Proc. of
the International Conference on Physics and Con-
trol – PhysCon 2003, 2003, St. Petersburg, Russia,
p.997-1002.
ОПТИМИЗАЦИЯ ДИНАМИКИ ЗАРЯЖЕННЫХ ЧАСТИЦ В УСКОРИТЕЛЕ
С ТРУБКАМИ ДРЕЙФА
Е.Д. Котина
Рассматривается проблема оптимизации динамики заряженных частиц в линейном ускорителе с трубка-
ми дрейфа. Предлагается дискретная модель оптимизации, основанная на уравнениях Соловьева. Для оцен-
ки динамики вводится функционал, позволяющий проводить совместную оптимизацию программного и воз-
мущенных движений с учетом плотности распределения частиц в фазовом пространстве. Выписывается ана-
литическое представление вариации предложенного функционала, дающее возможность построения направ-
ленных методов оптимизации.
ОПТИМІЗАЦІЯ ДИНАМІКИ ЗАРЯДЖЕНИХ ЧАСТОК У ПРИСКОРЮВАЧІ
З ТРУБКАМИ ДРЕЙФУ
Є.Д. Котіна
Розглядається проблема оптимізації динаміки заряджених часток у лінійному прискорювачі з трубками
дрейфу. Пропонується дискретна модель оптимізації, заснована на рівняннях Соловйова. Для оцінки
динаміки вводиться функціонал, що дозволяє проводити спільну оптимізацію програмного і збуреного рухів
з урахуванням густини розподілу часток у фазовому просторі. Виписується аналітичне зображення варіації
запропонованого функціонала, що дає можливість побудови спрямованих методів оптимізації.
130
1. Introduction
оптимизация динамики заряженных частиц в ускорителе
с трубками дрейфа
|