div class="t m0 x0 h7 y11d ff3 fs4 fc0 sc0 ls3 ws12">seconds, can force the plasma out of its operational lim-
its.
Artificial neural networks and support vector machines
(SVMs) are capable of building general models, by
“learning” from the data in the so-called “training proc-
ess”. In the paper [17] the complete evolution of the con-
sidered discharges is followed from the beginning to the
end to determine whether a disruption if fortcoming or
not. We will try to describe the using of artificial neural
networks in investigation of behavior of nonlinear Vlasov-
Poisson-Fokker-Planck systems that describe the turbu-
lence.
In the past decades, the fuzzy logic theory has been
shown to be an effective approach to dealing with the
analysis and synthesis problems of nonlinear systems.
Among various models available to fuzzy systems, the
Takagi-Sugeno (T-S) fuzzy model was a notible one
which was a linear system constructed to approximate a
nonlinear plant. Roughly speaking, the T-S fuzzy model
is a system described by fuzzy IF-THEN rules which can
give local linear representation of the nonlinear system
by decomposing the whole input space into several par-
tial fuzzy spaces and representing each output space with
a linear equation.
We consider a generalization of the traditional Ta-
kagi-Sugeno fuzzy system that includes both the pa-
rameter uncertainities and the stochastic disturbances.
The corresponding ith fuzzy rule is formulated in the
following form:
Plant rule i:
If is

1t
1i
and...
pt
ip
is
then

 
d
,,
ii i
ft




d ,
i
A
A fttftft

 
t t
 
1, 2,,i r
I
(6)

q
itR
are uncorrelated zero mean
Gaussian white noise processes with covariances .
q
There been performed studies on the correlation be-
tween electrostatic and magnetic fluctuations. The con-
nection between turbulent density fluctuations and low-
dimensional chaotic dynamics has been quantitatively
verified by using suitable numerical algorithms like the
correlation dimension and the statistical distribution of
recurrence times.
One of the most difficult challenges in computational
statistical physics is sampling of low-temperature equi-
librium states of the systems. Standard Markov Chains
Monte Carlo (MCMC) methods become stuck in local
minima and are unable to correctly sample the equilib-
rium distribution. In parallel tempering, many replicas of
the system are simultaneously simulated using a standard
MCMC method with each replica at a different tempera-
ture. Replica exchanges are allowed that permit replicas
to move between low and high temperatures. We must
use the adaptive Monte Carlo.
Competitive learning neural networks are regarded as
a powerful tool for online data clustering to represent a
non-stationary probability distribution with a fixed nu-
mer of weight vectors. One difficulty in applications of
competitive learning neural networks to online data clus-
tering is that most of them require heuristically-prede-
termined threshold parameters of balancing a trade-off
between accuracy, i.e. error minimization performance,
and speed of adaptation to the changes in source statistics.
Both quick adaptation and error minimization are simul-
teneously accomplished without any carefully predefined
parameters.
When modeling transport processes in tokamak plas-
mas the following assumption prevails: widely separated
regions of the plasma do not significantly interact with
each other. This paradigm states that any particle or heat
flux can accurately be described using a set of local
transport coefficients.
Transport events are random and accurately described
by a classical Gauss-Markov process. Nor is taken into
account the fact that a large localized event may have a
significant aftereffect in the same region it occured in.
Any transport model should endeavor to encompass the
possibility for a nonlocal, nondiffusive action at a dis-
tance. The heat flux Q(r, t) has been temporally averaged
over a collision time; we thus only addressed the spatial
question of the nonlocal, nondiffusive (non-Gauss-Markov)
behavior of Q. An equally question is now to address its
Markov (temporal) counterpart. A strong nonlocal, non-
diffusive spatial behavior is found, stremming from the
selforganized transport.
At least, the available data does not allow to conclude
on the possibility of non-Markovian (memory) effects.
With the introducing of artificial neural networks for the
non-Markovian process is transfered into Markovian
process [18].
Copyright © 2012 SciRes. JMP
D. RASTOVIC 1865
6. Drift Kinetic Equations and Advanced
Optimal Control
The analysis of electrostatic fluctuations is of great in-
terest in understanding the degradation in confinement of
a magnetically confined plasma. Diagnostic methods are
mostly based on digital correlation of data applying dif-
ferent models for their analysis. Some diagnostics are
based on a statistical processing of signals.
We must identify systematically and properly the kind
of instability responsible for the turbulence and the
anomalous transport. The anomalous particle transport,
as well as the correlation between the existence of an
instability and its contribution to the transport, is one of
the most important issues in understanding the confine-
ment of a magnetized plasma.
The most notable one is that toroidal rotation reverses
its direction after a low-mode ( L-mode) to H-mode tran-
sition. To explain this phenomenon, one has to deal with
the toroidal momentum equation directly. To this end, we
adopt the neoclassical quasilinear theory to calculate tor-
oidal stress induced by electrostatic fluctuations and ex-
amine its qualitative behavior. Of course, this neoclassi-
cal theory must take into account the fact that the pres-
ence of magnetic island breaks the toroidal symmetry of
a tokamak.
The linear drift kinetic equation with plasma flows as
the driving terms is







2
2
32
1
2123
25
E
t
2
M
f
vVnV fC
t
vvvv
VBBLq
 

 

f
B Bpf
v
(7)
where f is the perturbed particle distribution function,
is particle speed paralel to B, v is the particle speed,
E
V
EB
t
is the drift, E is the electric field, C is the test
particle part of the Coulomb collision operator, V is the
plasma flow velocity, q is the heat flow, is the ther-
mal speed of the particle,
v

32 2
152Lx is a Laquerre
polinomial, t
x
vv and
M
f
is the equilibrium shifted
Maxwellian distribution function [19].
In Ohmically heated tokamak plasmas or in RF-heated
tokamak plasmas where wave momentum is zero, there is
no obvious toroidal momentum source. The steady-state
toroidal rotation velocity profile is determined approxi-
mately by balancing the diffusion term and the convec-
tive term. There are many partial results of the applica-
tion of drift kinetic equation for different regimes of to-
kamak plasma confinements.
Many physical dynamical systems are spatially ex-
tended and exhibit cooperative wave patterns as well as
high-dimensional chaotic attractors. Example for this
kind of complex dynamics is interacting spiral waves in
excitable media.
Only some model parameters have to be determined
from experimental data. Often, however, even the struc-
ture of the models is not or only partially known and has
to be identified using observed data. Structure identifica-
tion and parameter estimation, several methods have
been suggested based on non-linear regression, state
space reconstruction, perturbation or synchronization.
More difficult to identify and to model are structural
inhomogeneties which occur due to long-range connec-
tions between different(remote) locations in the spatially
extended system introducing additional coupling. Correct
identification of these long-range links is of crucial im-
portance for correct modeling. A network is balanced if
the in-degree of any nodes is equal to its out-degree.
We shall suggest a system identification method based
on steady-state stabilization. To stabilize a steady state,
we add to control signal i such that we have con-
trolled network. Then we exploit the dependence of the
equilibrium on control parameters. In this way we obtain
all relevant information to identify the underlying dy-
namics as well as coupling structure [20].
u
We shall consider in the following time-spatially ex-
tended systems that can approximately be described by
large sets of partial differential equations. We first stabi-
lize a stationary state (at least in some region) and then
exploit the dependence of the equilibrium on control pa-
rameters. In this way we obtain all relevant information
to identify the underlying dynamics as well as coupling
structure.
Statistical behavior described by power-laws is ap-
pealing to attribute to the existence of “criticality” in the
underlying dynamics. In the statistical mechanics sense,
this means the proximity of a phase transition, and one of
the perminent ideas in this respect is “self-organized
criticality” (SOC). This is usually taken to mean that no
fine-tunning to a critical point in the classical sense is
needed and the system at hand is driven to such a critical
state.
The measured signal can be considered after e.g. spa-
tial or temporal thresholding. Example is given by the
local particle flux in the perimeter of a fusion device. In
particular for classical SOC systems, with uncorrelated
driving, global waiting time statistics are described most
often by a Poisson process.
Understanding the field lines chaotic structures in sta-
tionary flows can be considered as the first step towards
understanding transport in these flows. This chaos of
field lines has been investigated for a long time in plasma
physics, especially when concerned with the conception
of magnetically confining devices such as tokamaks.
Potential interest in computing field lines for a flux
free field is the possibility to compute Poincare sections
of the field lines. We should also be able to discriminate
Copyright © 2012 SciRes. JMP
D. RASTOVIC
1866
z
is the charge number of the particle species the physical importance of islands of regular motion
within a stochastic sea in the Poincare section. Let v be a
three-dimensional vector field. Field lines of v are curves
which are tangent to the field at any point.
,ie

v
, E is the poloidal electric field, b is direction
of magnetic field, Simulation and Control of Turbulence at Tokamaks with Artificial Intelligence Methods
Journal of Modern Physics, 2012, 3, 1858-1869
http://dx.doi.org/10.4236/jmp.2012.312234 Published Online December 2012 (http://www.SciRP.org/journal/jmp)
Simulation and Control of Turbulence at Tokamaks with
Artificial Intelligence Methods
Danilo Rastovic
Control Systems Group, Zagreb, Croatia
Email: dr.rastovic@excite.com
Received August 11, 2012; revised October 18, 2012; accepted October 28, 2012
ABSTRACT
The control of turbulence at tokamaks is very complex problem.The idea is to apply the fuzzy Markovian processes and
fuzzy Brownian motions as good approximation of general robust drift kinetic equation. It is obtained by using the
artificial neural networks for solving of appropriate advanced control problem. The proof of the appropriate theorem is
shown.
Keywords: Tokamak; Turbulence; Control; Artificial Intelligence; Drift Kinetic Equation
1. Introduction
High-ß regimes are difficult to operate in, mostly because
of macroscopic instabilities leading to the disruption of
the plasma column and termination of discharge. Whe-
ther driven by magnetic of electrostatic turbulence, anoma-
lous transport causes a large increase in the energy losses,
and poses severe constraint on the amount of auxiliary
power required to heat up the plasma to the high tem-
peratures required for thermonuclear ignition.
When the plasma rotates either toroidally or poloidally,
both the energy transport as well as the macroscopic sta-
bility improve significantly. The plasma rotation can be
either spontaneous or driven by neutral beam injection or
radio frequency (RF) heating.
Flow modified equilibria will almost certainly exibit
stability properties different from the static equilibria,
which need to be investigated with new kinds of non-
variational stability codes. Flow shear (both toroidal and
poloidal) is found to be stabilized, while the bulk toroidal
rotation can be either stabilizing or destabilizing.
Turbulence in plasmas is an ever present theme in
plasma physics, including in reactor-size tokamaks. An
obvious way to begin a study of the kinetics would be to
use an experimentally measured turbulence potential. The
second possible way is to simulate the generation of the
turbulence by a given system of equations. We can con-
sider all possible types of motion: regular motion, Gaus-
sian diffusion and Levy walk diffusion.
Pure averaging prevents one from recognizing impor-
tant effects, in particular the effect of “islands around
islands”. This type of potential region, which separates
regular motion from the “stochastic sea”, is considered
by Zaslavsky et al. [1] as the real seed of anomalous dif-
fusion. The problem is how to synchronize all these ef-
fects by the methods of artificial intelligence.
Using the advantage of using the Poincare surface-
of-sections, we can track the changes in the dynamics of
particles as the initial conditions approach the threshold
between regular motion and a stochastic sea (the stick-
ness effect), as well as the fractality of the kinetics.
In view of significance of the control of delayed neural
networks, in recent years, the stabilization and synchro-
nization of neural networks with delays have been exten-
sively investigated by many researchers [2]. As is widely
known, noises do exist in a neural network due to ran-
dom fluctuations and probabilistic causes in the network.
Thus, it is necessary and rewarding to study stochastic
effects to the dynamic behaviours of neural networks [3].
Operations with high-ß plasma are advantageous in the
tokamak reactor from the economical point of view. The
high-ß operations mean a high fusion output even with a
compact device, that is, the operation can be said to be
economically efficient. On realistic devices, the stabiliz-
ing effect is lost owing to the wall resistivity since a
conducting wall has a finite resistivity. The finite resis-
tivity of the conducting wall induces another instability,
that is, the resistive wall mode (RWM).
Since the RWM is thought to limit the achievable ß in
a realistic device, RWM stabilization is one of the key
issues for ITER and reactors. Theoretically, plasma rota-
tion suggested to strongly affect the RWM stability.
Experimentally, plasma rotation stabilization has been
observed. For understanding the stabilization with a slow
plasma rotation, kinetic effects of bulk plasma and fast
C
opyright © 2012 SciRes. JMP
D. RASTOVIC 1859
ions have been taken into account. From these reasons
we will apply advanced variational calculus which in-
clude also the applications of artificial neural networks in
the process of learning during optimization of results.
2. Tokamak Experiments
For more than 30 years fusion researchers have sought to
understand a variety of abrupt transitions in plasma oper-
ating conditions observed in tokamak experiments, and
to anticipate others that might be present in fusion reac-
tors, by analyzing instabilites in the coupled plasma par-
ticle, momentum and energy balance—i.e. thermal insta-
bilities.
A strong temperature dependence of energy transport
loss rate has been identified a driver or a stabilizer of
thermal instabilities, depending on the sign of the tem-
perature dependence. Many of these drivers of thermal
instabilities are primary edge phenomena. Because the
onset of the edge-related thermal instabilities can be
theoretically related to local or average densities exceed-
ing a threshold value, density limits for the onset of these
observed phenomena can be related to the onset of the
respective thermal instabilities. The onset conditions are
determined by a balance between destabilizing term-
impurity and the stabilizing term-conduction.
The temperature dependence of the threshold densities
is more complex, because of the very different tempera-
ture dependence of the various stabilizing and destabi-
lizing terms.
The thermal instability explanation of the L-H and
H-L transitions and of observed confinement degradation,
while supported by experimental comparations, is new,
and there exists a much larger body of competing expla-
nations for these particular experimental phenomena.
Looking ahead to burning plasmas, the role of the de-
stabilizing temperature dependence of the fusion cross
section is well understood, several active control mecha-
nisms have been investigated, and the possibility of in-
herent stabilization via the temperature dependence of
thermal transport coefficients has been identified.
Magnetic islands and neoclassical tearing modes (NTMs)
can produce sawtooth crashes and disruptions. Many of
the proposed solutions are related to the employment of
resonant magnetic perturbations (RMPs). A RMP is a
magnetic perturbation characterized by a helical resonant
inside the plasma and typically produced by a dedicated
set of active coils [4]. Recently, RMPs have been utilized
to produce stochasticity at the plasma edge.
We can show the equivalence of the first variation of
the electrostatic entropy and the canonically averaged
Lagrangian with respect to adiabatic variations. One
finds equivalence between the Hamilton’s action princi-
ple and the vanishing of the first variation of nonlinear
electrostatic entropy with respect to adiabatic variations.
The magnetic configuration of the plasma is time in-
dependent. This is the case of the tokamak in static equi-
librium. Time independent magnetic configurations of
the plasma associated with a time independent magnetic
entropy are called “plasma states”. The electric field pre-
sent in a plasma state can only be the stationary electric
field induced externally.
The requirement that the magnetic entropy is station-
ary in the tokamak implies the balance between the net
power deposited on electrons and the power lost by elec-
tron thermal conductivity. The additional requirement
that the balance holds locally leads to an equation for the
profile of the current density induced by external electric
field (the stationary magnetic entropy equation) [5]. We
can investigate how the magnetic entropy changes in
time when the plasma evolves in accordance with Max-
well equations.
One can discuss the stability of the magnetic equilib-
rium in a variety of physical situations depending on the
special form of the current density. The pressure profiles
in tokamaks are restricted by the requirement that the
thermal energy of the plasma be minimum for fixed
magnetic entropy and fixed total current.
Chaotic systems have been exploited for applications
such as control and random number generation. In these
applications, it is desirable that the underlying chaotic
attractor is robust so that random fluctuations or small
disturbances would not cause the system to transit to a
periodic state.
There are many problems faced, including the effect of
injection heating and the different fuelling scenarios for
power and particle balance of fusion plasma. We can
consider not only fluctuations but also modify the time
variation of particle fuelling scenarios. It is shown that a
slow variation of the helium ash density with time can be
used for operations path-changing in deuterium-tritium
fusion plasma. The main target of the study is the opti-
mization of the plasma operation scenario in future fu-
sion reactors including ITER.
The evolution of equilibrium was considered in con-
nection with the diffusion of magnetic fluxes and the
redistribution of the current, without taking account of
the evolution of the pressure. Here attention is concen-
trate here on the correlation of the evolution of equilib-
rium and heat transfer, and the current distribution is
taken to be specified.
The attainment of thermonuclear regimes, with high
temperature, very long resistive diffusion time scales,
high
p
effects, is finally dependent on major problem
of tokamak physics and operation, namely the control of
the inevitable instability of slow-growing resistive modes.
In many cases actual plasmas are far from thermal equi-
librium, being inhomogeneous or having non-Maxwel-
Copyright © 2012 SciRes. JMP
D. RASTOVIC
1860
lian velocity distributions, features which significantly
complicate the analysis of experimental data. In addition,
these departures from thermal equilibrium can trigger
instabilities, whose growth and non-linear saturation even-
tually lead to the onset of turbulence.
The power of the thermonuclear burning will be stabi-
lized at a fixed level by automatic regulation of the
power of additional heating. In this case the reactor oper-
ates in the subcritical regime, i.e., the parameter of the
plasma are chosen so that the power of the thermonuclear
reactions is slightly less than the power lost. This differ-
ence is compensated by additional heating, the power of
which follows the thermonuclear power via negative
feedback.
It is obvious that as the thermonuclear power increases
feedback will increase the additional heating power.
Therefore the installation must be provided with reserve
power capable of compensating the oscillations of S
for α-particles. The value of this reserve power depends
on the parameters of the plasma and the technically de-
termined desirable accuracy of the power regulation.
In fusion reactor design, a radiative divertor concept is
widely adopted to suppress erosion at the divertor plates
and to reduce the localized heat load on the divertor
plates.
3. Tokamaks and Turbulence
In the coming year we plan to contribute broadly to
theoretical modelling of ITER-like and spherical toka-
maks. The range of topics and activities envisaged is
listed below:
Contribute to projects such as equilibrium reconstruc-
tion, fast particle physics and turbulence code bench-
marking.
Develop and implement the fully parallelised version
of global two-fluid turbulence simulation code with
demonstration applications to MAST and other toka-
maks.
Use the local gyro-kinetic code for modelling micro-
instabilities in MAST.
Continue the modelling of the effect of tearing mode
islands on fast particle confinement in spherical to-
kamak geometry with application to MAST.
Understanding the role of turbulent processes in mag-
netically confined plasmas is central to understanding
their transport and confinement properties.
We can analyze turbulent electrostatic field in a sta-
tionary toroidal magnetoplasma, created by radio-fre-
quency waves and confined by two different toroidal
magnetic fields. Observations suggest that turbulence has
recurrent properties, as those observed in recurrent fully
chaotic low-dimensional systems. Therefore, evolution of
measurements of low-dimensional systems can be used
to describe the recurrence observed in the tokamak edge
turbulence.
The investigation of the correlations between, on the
one hand, the occurence of transport barriers and im-
proved confinement in magnetically confined plasmas,
and on the other hand, electric fields, modifed magnetic
shear and electrostatic and magnetic turbulent fluctua-
tions necessitates the use of various active means to ex-
ternally control plasma transport. It also requires one to
characterize fluctuations of various transport barriers and
to elucidate the role of turbulence driving and damping
mechanisms, including the role of the plasma edge prop-
erties.
There are some conventional control theory methods
of the stabilization of the resistive wall mode (RWM)
instability at tokamaks [6]. Our aim is the application of
the conventional control theory together with the meth-
ods of fuzzy neural control theory.
The most challenging aspect of the H-mode physics is
the sudden disappearance of the turbulence whereas its
driving forces—the gradients at the plasma edge-increase.
The turbulence, which normally limits the confinement,
induces a sheared plasma flow, which acts back to its
generating origin and quenches the fluctuations. One part
of this process-turbulence produces zonal flows via Rey-
nolds stress plays a role in normal fluids also.
The turbulence tends to drive large scale flows, which
are in turn mitigated through interaction with toroidal
magnetic equilibrium. The understanding of turbulent
transport processes which govern the energy, momentum
and current distributions in tokamak plasmas is important
tool to optimizing the economically viable design of fu-
ture power plants based on tokamak concept. Zonal flows
can lead to substantial reduction of turbulent transport in
localized region as transport barriers.
Understanding and predicting turbulent transport is a
key issue on the way toward commercially, viable fusion
reactors. Indeed turbulence control the confinement con-
trol of any magnetically confined plasma.
Obviously, gyrokinetic theory is intimately entangled
in the physics of turbulent transport in magnetized fusion
plasmas. Zonal flows are generated by turbulence and
back react on turbulence via vortex shearing and convec-
tion from locally stable to unstable regions. We obtain a
familiar expression of the local gyrokinetic equation


2
0
0
0
1
where .
s
ss
s
MS
GY ssMS
ssMS s
c
vbbemvbbBh
teB
fc
Cfeb f
tB
fHf h


 
 







(1)
This model is an open system developed to study local
quasi-steady turbulent transport with fixed background
Copyright © 2012 SciRes. JMP
D. RASTOVIC 1861
profiles. Although the conservation properties in the ori-
ginal gyrokinetic equations are lost, the quality of the
nonlinear simulation is often tested using the entropy
balance relation [7].
Domain decomposition is in principle applicable to the
all three classes of schemes (Lagrangian, Eulerian and
semi-Lagrangian). The geometry of the magnetic con-
figuration strongly effects microinstabilities and turbu-
lence. The idea is the application of fuzzy neural simula-
tions of the Vlasov-Poisson-Fokker-Planck (VPFP) equa-
tion to obtain some kind of variational calculus for the
plasma confinement problem in fusion magnetic devices.
The stochastic part of kinetic equation is described with
Brownian motion. The Markov processes are good ap-
proximations of Brownian motion. In tokamak physics is
necessary work with interval numbers as uncertainities.
We can interpret these intervals with words. In this case
we obtain fuzzy Markovian processes and fuzzy Brow-
nian motions. The new changes are incorporated in ki-
netic plasma equation as fuzzy neural networks.
Non-equilibrium turbulent processes are anisotropic,
non-local, multi-scale and multi-phase, and often are
driven by shocks or acceleration. Their scaling, spectral
and invariant properties differ substantially from those of
classical Kolmogorov turbulence. At atomistic and meso-
scales, the non-equilibrium dynamics depart dramatically
from a standard scenario given by Gibbs statistical en-
semble average and quasi-static Boltzmann equation.
At dissipative scales, where the fluid flow is differen-
tiable, the phase-space density of particles is supported
on a dynamically evolving fractal set. This attractor is
characterized by non-trivial multiscaling properties.
Description of transition from regular to random re-
gimes of motion and characterization of the intermediate
states where both regular and random wave motions are
present and mutual inter-connected, is an intriguing and
challenging problem. Recent technical progress allows to
investigate, both experimentally and numerically, the
correlation properties of fully developed turbulence in
terms of particle trajectories.
It has recently become possible to compute precise
equilibrium, traveling wave, and periodic orbit solution
above the onset of turbulence. These invariant solutions
capture the complex dynamics of unstable coherent struc-
tures in wall-bounded flows and provide a framework for
understanding a turbulent flows as dynamical systems.
We present a number of weakly unstable equilibria and
periodic orbits and visualizations of their physical and
state-space dynamics. Applications could be found in
magnetic confinement fusion.
The understanding and reduction of turbulence trans-
port in magnetic confinement devices is not only an aca-
demic task but also a matter of practical interest, since
high confinement has been chosen as the regime for
ITER and possible future reactors because it reduces size
and cost. Turbulence comes in two classes: electrostatic
and magnetic turbulence. Turbulence can be externally
controlled, which led to better and better confinement.
Electrostatic turbulence stabilization concept has the
universality, needed to explain ion transport barriers at
different radii seen in limiter and divertor tokamaks with
a variety of edge biasing schemes. Magnetic turbulence
drives the anomalous electron heat conduction [8].
Turbulence plays a very important role in particles and
energy cross-field transport to the wall in the edge
plasma. The problem is extract the maximum of informa-
tion from data. Taking into account the intrinsic inter-
mittent and nonlinear character of turbulent data, the
Fourier methods are not well suited. The useful can be
the advanteges of the wavelet techniques and how these
techniques make it to resolve problem of quantitative
characterization of intermittency and of the self-similar-
ity properties of turbulence and of space-time characteri-
zation by measuring correlations between data taken
from different locations.
Theoretical description of non-equilibrium transports
is a challeging problem due to singular aspects of the
governing equations. Furthemore these processes are sto-
chastically unsteady and their fluctuating quantities are
essentially time-dependent and non-Gaussian. It is shown
that their invariant, spectral, scaling and statistical prop-
erties differ substantially from those of isotropic homo-
geneous turbulence.
4. Approach with Control Theory
Some characteristics of transport in fusion plasmas indi-
cate that it may be a manifestation of a self-organized
and critical system (SOC). These characteristics are: ex-
istence of critical gradients, the existence of transport
events that do not obey diffusive equations and global
scaling. If a magnetically confined plasma is a SOC sys-
tem it might lead to novel ways of controlling turbulence.
The Hurst coefficient is closely related to the decay of
the auto-correlation function at long lags and implies the
existence of long-range time correlations. The range of
values of H in the edge plasma is between 0.6 and 0.74
but in the scrape-off-layer (SOL) displays a much larger
variation, between 0.5 and 1.
At W7-AS, a radial profile of H was obtained that
proved to be reproducible in various similar discharges
having a maximum near separatrix [9]. The process of
autonomous convergence towards an ordered complex
form, also called self-organization process, requires form
that have already reached a somewhat advanced stage in
ordering process. For example, in nonlinear thermody-
namics this principle will be called self-organization. In
the selforganization theory—a convergence towards the
Copyright © 2012 SciRes. JMP
D. RASTOVIC
1862
maximum interaction is obtained (positive feedback [10]).
High-power radio frequency in the ion-cyclotron range
of frequencies (ICRF) have the potential to heat and con-
trol the ITER plasma through localized energy depo-
sition, driven current, and driven plasma flows. Fokker-
Planck coupled radio-frequency simulations show that
because of the high plasma density, energetic ion tail
formation in ITER is typically weak, with the expection
of the minority deuterium heating scheme where strong
tails can develop on the minority ion distribution.
Control of plasma density and temperature magnitudes,
as well as their profiles, are among the most fundamental
problems in fusion reactors. In thermally unstable zone,
and active control system is necessary to stabilize the
thermonuclear reactions. The controller makes use si-
multaneously of the modulation of auxiliary power, the
modulation of fueling rate and the controlled injection of
impurities as actuators. Radiative divertor conditions
were obtained by injection of gaseous impurities into
divertor.
When the dynamics within a scattering region is cha-
otic, we say that we have a chaotic scattering. Chaotic
scattering is a very active research topic in dynamical
systems. In analogy to more general dynamics, chaotic
scattering can be hyperbolic or nonhyperbolic. Hyper-
bolic scattering is associated with an exponential decay
of probability of particles to be found in the scattering
region after a given time t. In the paper [11] is given scat-
tering theory for linear transport for the case of neutrons.
The exponential decay of probability for particles in the
case of linear transport theory for neutrons and ions is
shown in the papers [12,13].
Nonhyperbolic scattering is characterized by the pres-
ence of Kolmogorov-Arnold-Moser (KAM) islands in
phase space. These islands surround marginally stable
periodic orbits. Nonhyperbolic escape is characterized by
a power law distribution of probability [14]. This is the
case of general transport theory for ions. The interesting
question is when the power law converges in averaging
to the exponential decay. There is mathematical inter-
prettation of such assertion [15].
New simulation tools open the route to the investiga-
tion of plasma turbulent transport with increasingly self-
consistent interplay between various scales and transport
mechanisms to drive such a complex system.
The system is governed by the following dimen-
sionless equations:
 
()
G
EgG
GG
d
d
GG
G
G
v
f
f
tv
 
f
t
SC
f Df
 
 
vvv
(2)
[16]. The guiding-centre distribution function G
is
normalized as follows:
00T
3
GG
f
nv f. The transverse
drifts and parallel dynamics are governed by the following
set of equations:

2
EJ
B

B
v

2
2
1
gG
vB B
BB

B
v

d
d
G
GE
vB
JBv
tB


v (3)
Equation (2) conserves the three equilibrium motion
invariants in the collisionless regime, namely the mag-
netic momentum
(trivial), the total energy

22,,
G
HvBJ r


LRv
(4)
and the toroidal kinetic momentum G

 due
to axi-symmetry (with
the poloidal magnetic flux).
Ion temperature gradient (ITG) turbulence is often
found to be main cause of anomalous transport in fusion
experiments. If the plasma conditions are chosen appro-
priately, it is also possible to experimentally investigate
“pure” trapped electron mode (TEM) regimes and com-
pare with turbulence simulations. But in more conven-
tional situations, the application of several heating
methods (ECRH) plus neutral beam injection (NBI) and/
or a sufficiently high collisionality leads to finite (and
often similar) values for the temperature gradients of all
species, which usually means that several linear instabili-
ties are present simultaneously.
The interaction between the different unstable modes
at the TEM-ITG transition produces some interesting
phenomena, including a supression mechanism for the
particle flux. Some of these features can be understood at
least qualitatively by quasilinear investigations, which
offer the possibility of extended parameter scans.
The collective configuration can exchange electric
charges with the medium and therefore the amplitude of
the collective quantities is not fixed but fluctuates. In the
case of an inhomogeneous plasma one can define an
electrostatic entropy functional which is sensible to the
local structure of the equilibrium. The electrostatic inter-
action energy has a precise physical meaning.
The simple case of the electrostatic equilibrium is de-
scribed by the one-dimensional Vlasov equation, static in
the laboratory frame of reference

0
ss s
fq f
vEx
xm v

 

(5)
An oscillatory one-dimensional static solution of the
electrostatic Vlasov equation is always associated with a
non-Maxwellian distribution function. It can be shown
that the minumum of the electrostatic entropy corre-
sponds to the instability of the collective equilibrium
predicted by Vlasov equation. Any distribution function
Copyright © 2012 SciRes. JMP
D. RASTOVIC 1863
with a single maximum is stable and corresponds to the
maximum of the entropy. The Maxwellian case is con-
tained consistently in the theory.
The ultimate application of fusion power as an energy
source requires not only a high fusion reaction rate but
also steady-state operations, which often necessitates high
plasma pressure. Unstable resistive-wall modes (RWMs)
were found to have been suppressed by either rapid
plasma rotation or active feedback control in various
toroidal devices. To study the details of the RWM feed-
back process and assess the RWM feedback performance,
highly reproducible Ohmic discharges with fast plasma
current ramp-up unstable to the current-driven RWMs at
edge safety factor have been developed in the DIII-D
tokamak.
The error fields are externally imposed non-axisym-
metric fields that couple to the kink mode and arise from
inevitable irregularities of the magnetic configuration of
the tokamak. There is a dynamically varying error field
whose origin is unknown and appears only during plasma
operations. From this point of view necessity is to have
the real time controller in the process. Not all the feed-
back coil currents are directly used for suppressing the
RWM. In that regard, it has been challenging to identify
the partition of coil currents directly.
The direction of the toroidal rotation can be reversed
when the mode frequency decreases. This provides a
natural explanation for the toroidal flow reversal when
plasma make a L-H transition observed in Alcator C-
MOD. The quasilinear theory of toroidal momentum
confinement is not only applicable to Ohmic-heated
plasmas, but also to radio-frequency-wave (RF) heated
tokamak if there is no toroidal momentum input associ-
ated with the RF sources. In this regard, the theory may
provide an explanation for the “corotation” observed in
RF-heated Alcator C-MOD plasmas. One can speculate
that either RF wave electric field stabilizes ion tempera-
ture gradient driven turbulence or RF heating modifies
ion temperature profile and causes mode frequency to
decrease. This decrease leads to “corotation”. The dif-
ference in RF-induced
E
B drift can cause a charge
accumulation opposite to that of the instability. The in-
stability could thus be stabilized. The stabilization me-
chanism can be due to the finite ion Larmor radius ef-
fects.
Now, after some 30 years of tests using the Alcator se-
ries of reactors, the MIT researches have found one mode
of operation, which they call I-mode (for improved) on
which the heat stays tightly confined but the particles,
including contaminants, can leak away. When a fusion
reactor operates, the impurities accumulate. The operat-
ing conditions and the control requirements to stay in
I-mode need to be better undestood.
The I-mode eliminates or greatly reduces the occur-
ance of unpredictable bursts of heat from the edge to the
confined plasma. Depending of how the strenght and
shape of the magnetic field are set, both heat and particles
can constantly leak out of plasma (in a setup called L-
mode, for low-confinement) or can be held more tightly
in place (called H-mode, for high confinement). In I-
mode the heat stays tightly confined, but the particles,
including contaminants, can leak away. The findings
could be useful when fusion power is sustained mostly
by “self-heating” and don’t need large amounts of out-
side power.
Experimental results indicate the existence of azi-
muthal (toroidal) plasma rotation in tokamaks subjected
to neutral beam heating. In field-reversed configurations
azimuthal rotation is responsible for a type of instability
that may destroy plasma confinement. One possible ap-
proach to investigate the effects of rotation on MHD
equilibrium and stability properties would be to obtain
numerical solutions of the 3-dimensional ideal MHD
equations. Magnetic flux surfaces rotate rigidly with the
plasma, according to Alfen’s theorem, and are character-
ized by poloidal flux and current functions that satisfy an
elliptic partial differential equation which, in the limit of
vanishing rotation, is reduced to the Grad-Shafranov
equation. For these purposes is good apply the approach
with fuzzy mathematics. In this case, the fuzzy Mark-
ovian processes describe the behavior of the solution of
Fokker-Planck equation, as generalization of Brownian
motion, for Vlasov-Poisson-Fokker-Planck equation.
5. The Methods of Artificial Intelligence
Stellarator posses an equilibrium parameter space that is
greather than that of tokamaks because of the full three-
dimensional nature of stellarator plasmas. Some disrup-
tive instabilities nowdays can be avoided thanks to the
increased experimental experience with tokamak opera-
tion, and instabilities of the plasma position are tackled
with improved magnetic control techniques, for instance.
Plasmas with shaped cross sections have higher mag-
netohydrodynamic stability limits, but require sophisti-
cated feedback control. Moreover, vertically elongated
plasma is vertically unstable and requires feedback sta-
bilization. We can consider the design of a Kalman filter
to reconstruct the states relevant for plasma vertical sta-
bilizator. But for general nonlinear, stochastic Vlasov-
Poisson-Fokker-Planck systems we must use Ito calculus.
Genetic algorithms rely on the concept of natural se-
lection mechanisms and genetic operators. The GAs is
generally decoposed into the following steps:
1) Random construction of an initial population and
evolution of its fitness.
2) Choice of the subset according to its fitness and
breading a new population.
Copyright © 2012 SciRes. JMP
D. RASTOVIC
1864
3) Evaluation of the fitness of each new individual and
replacement of the old population with the new one.
4) Checking whether the fitness has reached a prede-
fined value or the iteration number (generation) has
reached a certain value. If not, reiteration from step 2.
This is the way toward application of modern ad-
vanced Monte Carlo methods.
Disruptions are sudden and unavoidable losses of con-
finement that may put at risk the integrity of a tokamak.
However, the physical phenomena leading to disruptions
are very complex and non-linear and therefore no satis-
factory model has been devised so far either for their
avoidance or their prediction. For this reason, machine
learning techniques have been extensively pursued in the
last years. Disruptions can be triggered by various insta-
bilities which, on time scale even of the order of milli-
is the parallel velocity of particle,
and C and Q are the linearized Coulomb collision opera-
tor and the RF quasilinear operator, correspondingly. The anomalous transport is non-Fickian transport.
The entropy of nonequilibrium stationary states differs
from the entropy of local equilibrium states. In some
cases the long range correlations of the nonequilibrium
state is not captured by the Gibbs-Shannon entropy of
local equilibrium states. We can examine the entropy of
the stationary nonequilibrium states ,
N
[21]. We
solve the nonlocal-in-time transport equation with a

,
s
t
containing a power-law dependence in both s (a
Levy-like distribution) and t, which necessitates the
strong s, t coupling. The interplay between two coupled
power laws is clearly shown in the changes in the arrival
times and dispersion.
We write the drift kinetic equation as

ˆE
k
PB 0
f
qf
tmv
f
v

ev v vf
v
v
qf
mBv
 


 




 

(8)
where 22vB
v EB
is the magnetic moment.
Here the E term is the nonlinearity. It
dominates and is responsible for the first saturation of an
instability. This is true both in fluid and in kinetic sys-
tems. The third term is the parallel nonlinearity. It gives
nonlinear Landau damping and has sometimes been ig-
nored in kinetic codes. The last term is the nonlinear
magnetic drift resonance. It is analogous to nonlinear
Landau damping but applies to the perpendicular mag-
netic drift resonance. It has sometimes been neglected in
kinetic codes.
f
Gyro-Landau fluid (or gyrofluid) equations are an ex-
tension of the usual equation to include models of kinetic
resonances and gyro-averaging which play important
roles in fusion plasma turbulence.
Based on the linearized kinetic drift equation and the
fluid equations including the density perturbation term
driven by RF waves, the criterion of the destabilization
of the poloidal plasma rotation is analytically derived. It
is shown that the edge poloidal plasma rotation can be
destabilized during electron cyclotron resonant heating
(ECRH) in the present RF power level [22].
The production of the poloidal density asymmetry by
using ECRH in the collisional plasmas is studied based
on the linearized drift kinetic equation:


0
10
f
fQ f
0
1
f
vbfz eEbvC
tW

 

 , (9)
where 0
We formulate the criterion of the destabilization of the
poloidal plasma rotation during ECRH in the collision
plasmas, which is relevant to the edge plasma in toka-
maks.
Calculation of the thermodynamic fluctuations can be
applied to the large class of the so-called “reactive” in-
stabilities. This class includes unstable modes important
for thermonuclear machines, as the ion or electron tem-
perature gradient modes (ITG) or (ETG). The relation
between the electrostatic entropy and the Lagrangian
description of motion of the high temperature system of
Coulomb interacting particles is investigated. We can
proceed to the derivation of the equation of the con-
strained motion according to the Lagrangian. The appli-
cation of the Hamiltonian’s action principle gives
f
and 1
f
are the undisturbed and distorted
parts of electron distribution function due to rf waves,

22
11
1
d
dd0
d
p
tt
N
n
nn
tt
LL
Ltx tt
xt



 



 (10)
One obtains the equation of motion for the generic
s-particle.
The toroidal axis-symmetric equilibrium is described
by the Grad-Shafranov equation

2
4π
c
jRR
 (11)
The average of the Lagrangian can be expressed as a
linear function of the magnetic entropy and of the plasma
thermal energy only. L takes on the form

2
int
2
3p
mn V
LUV S
 
(12)
In condition of stationary equilibrium the thermal en-
ergy of the electrons is constant in time and the power
balance is expressed by the relation
ddd0
hext pE
qSEjp
 
  
 
q
(13)
Here h is the heat flux related to the thermal con-
ductivity of the electrons and the third therm contains the
contribution of the net power density deposited on elec-
trons (auxiliary power, electron-ion energy transfer, non
diffusive losses) in addition to the Ohmic contribution
described by the second therm.
Given the need for robust variational principles pre-
dicting the overall behavior of fusion plasmas, the en-
tropy-based methods are expected to be of increasing
importance in this field. The multiregion-relaxation va-
riational approach holds strong promise of being the
most satisfactory mathematical fundation on which to
Copyright © 2012 SciRes. JMP
D. RASTOVIC 1867
base the selection of the MHD toroidal equilibrium
problem posed by Grad over forty years ago. In the paper
[23] is studied an optimal control problem arising in
plasma transport which is governed by a singularly per-
turbed system.
A formulation of the Euler-Lagrange equations was
given for problems of the calculus of variations with
fractional derivatives. There is no general formulation of
a fractional version of Pontryagin’s Maximum Principle.
With a fractional notion of Pontryagin extremal, one can
try to extend it to the more general context of the frac-
tional optimal control [24]. The drift kinetic equation can
be derived from full kinetic equation, i.e. Vlasov-Pois-
son-Fokker-Planck equation.
We have the next theorem.
Theorem. Let us have the fusion reactor problem.
Then simulation and optimization of the results could be
obtained by the methods of advanced optimal control.
Proof. We consider the control systems of the Vlasov-
Poisson-Fokker-Planck equations
11
,,
f
Af BuuFfw
 wFfw 
,1,2,,
(14)
where A is direct sum of operators i
A
ik
 

Ev
 

,
iix iv
Av BF 
fff f
(15)
and 1
BF vv
,1
fff, vector f is direct sum of vec-
tors i [25]. ,2,,ik
,Etx is the force field
acting on particles. The strictly positive parameters
and
model a certain type of interaction between par-
ticles.
Consider the following IF-THEN rules:
IF is

1
m1i
M
and … IF

p
mip
is
M
THEN


 
dd
1, 2,,
ii di
 
,
f
tAEftAf
ik
 
thBut (16)
where
ij
M
are fuzzy sets and

1,,
p
mm are given
premise variables, and i are the uncertain matrices.
The fuzzy system is hence given by the sum of equations
E



 
,dd
1,2,,
ii di
i
f
taAEftAf
ik
 
thBut
a
 
d
T
(17)
where

i are the fuzzy basis functions [26]. We have
the general formalism of optimal control in the presence
of noise. Optimal feedback design based on deterministic
model is minimization of total fluctuation energy of in-
stabilities, as well as minimization of control power

 
0
f
tT
fNN
J
tftQft
utRutt
 
 
TT
NN
ut Rut
t ft

(18)
For Lagrangian


 
,
NN N N
T
Lfff Qf t
tAft BCu

 
 (19)
It holds that d0
dNN
LL
tf f





 


1T
utRBCt

 
with optimal control
.
Here
is activation function of a deterministic re-
current ANN. It can be calculated with Paricle in Cell
(PIC) method.
The general appropriate reduced model is
N
NN
fAfBCuDF

 (20)
N
DF
is stochastic noise, and where
is ac-
tivation function of a stochastic recurrent ANN [27]. It
can be calculated with Monte Carlo method.
The solution under optimal feedback is

NcN N
ABKf tD Ft

 
f
(21)
and

 
0
00
,
,d
NN
t
N
t
fttt ft
tDF


(22)
On this way the theorem is shown.
As actuators in the process we can use also NBI, ICRF,
ECCD and it must be used in such a way that the optimal
result is obtained.
We considered the following VPFP system with neural
networks with delays





1
1
,,
,,
i
ii ijji
j
ij jiji
j
f
A
fBCu wqftxv
t
hlf ttxvJ
 
 
(23)
,,, ,
12 k
ff f

,, i
f
f
where txv is the state of the
plasma at time t,
j
l

tt
w
denotes the activation function of
the ith neuron at time j. On the synchronization
problems of neural networks, we see that there have been
some research work [28]. We want to obtain and adap-
tive synchronization controller to achieve exponential
synchronization for a class of VPFP systems. Here ij
denotes the strength of the jth neuron on the ith neuron at
time t and in the space
,
x
vh

tt
, ij strength of the jth
neuron on the ith neuron at time j and in the
space
,
x
v,
jt
 
are time-varying delays of neural
networks.
The “self-organized criticality” (SOC) is usually taken
to mean that the system is driven to a critical state. We in-
troduce the following. The entropy of fuzzy dynamical
system has been defined. Let

1,2,,PA AAn
be a measurable fuzzy partition of X. The entropy
P

of P is defined as
H
H
P is a sum of n elements
of the type:


logmAimAi. The entropy of
fuzzy dynamical system
,,,
X
FmT is the number
Copyright © 2012 SciRes. JMP
D. RASTOVIC
1868
hT
 
sup ,H PT
 
hT
,
defined as , where the su-
premum is taken over all finite measurable fuzzy par-
titions of X and T:X to X is a fuzzy measure-preserving
transformation.
H
PT

,
is a limit, as n tends to in-
finity, of the expression H over span of potentions of T
multilied by P and then divided by n. The limit
H
PT exists for each fuzzy process with finite
measure. The entropy

,
H
PT was introduced by Kol-
mogorov and Sinai and is often referred to as the Kol-
mogorov-Sinai invariant. The Kolmogorov-Sinai invari-
ant was the first invariant which was not a spectral nature.
In the deterministic case in equilibrium state the mini-
mum of fuzzy Kolmogorov-Sinai entropy must be ob-
tained. If we consider the stochastic case then in equi-
librium the maximum of fuzzy Jaynes entropy must be
obtained [29].
In the case of plasma confinement there are often only
experimental obtained data that are not calculated by
formulas. It is the case when disruption is possible. In
these cases we must use the method of artificial neural
networks for describing the problem [30]. After imple-
mentation of artificial neural networks our advanced
variational calculus can be used to find the optimal path
of plasma transport without disruption.
7. Conclusion
The advanced optimal control of fusion plasma con-
finement is solved by the methods of artificial intelli-
gence and drift kinetic equations. It means that fuzzy
systems and artificial neural networks should be applied
in the process of optimization. On such a way the using
of neutral beam injection, ion-cyclotron range of fre-
quencies and electron cyclotron current drive can be ana-
lysed as actuators together with Vlasov-Poisson-Fokker-
Planck equations. We construct an asymptotic solution of
the equation of magnetohydrodynamics (MHD) at high
Reynolds numbers. One of the quantities that may be
considered in this context is the waiting time of an ex-
perimental or observational parameter. As numerical tools
we can use fuzzy, adaptive and local Monte Carlo meth-
ods or particle in cell method via advanced variational
calculus.
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