div class="t m0 x0 h7 y11d ff3 fs4 fc0 sc0 ls3 ws12">seconds, can force the plasma out of its operational lim-
D. RASTOVIC 1867
base the selection of the MHD toroidal equilibrium
problem posed by Grad over forty years ago. In the paper
 is studied an optimal control problem arising in
plasma transport which is governed by a singularly per-
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 . The drift kinetic equation can
be derived from full kinetic equation, i.e. Vlasov-Pois-
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-
where A is direct sum of operators i
fff, vector f is direct sum of vec-
tors i . ,2,,ik
,Etx is the force field
acting on particles. The strictly positive parameters
model a certain type of interaction between par-
Consider the following IF-THEN rules:
and … IF
are fuzzy sets and
mm are given
premise variables, and i are the uncertain matrices.
The fuzzy system is hence given by the sum of equations
i are the fuzzy basis functions . 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
NN N N
Lfff Qf t
It holds that d0
with optimal control
is activation function of a deterministic re-
current ANN. It can be calculated with Paricle in Cell
The general appropriate reduced model is
is stochastic noise, and where
tivation function of a stochastic recurrent ANN . It
can be calculated with Monte Carlo method.
The solution under optimal feedback is
ABKf tD Ft
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
where txv is the state of the
plasma at time t,
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 . 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
, ij strength of the jth
neuron on the ith neuron at time j and in the
are time-varying delays of neural
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
be a measurable fuzzy partition of X. The entropy
of P is defined as
P is a sum of n elements
of the type:
logmAimAi. The entropy of
fuzzy dynamical system
FmT is the number
Copyright © 2012 SciRes. JMP
sup ,H PT
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
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
PT exists for each fuzzy process with finite
measure. The entropy
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
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 . After imple-
mentation of artificial neural networks our advanced
variational calculus can be used to find the optimal path
of plasma transport without disruption.
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
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