## The algorithms of the program control construction for some classes of the dynamic systems

#### Citation

Chalykh, E. (2009). The algorithms of the program control construction for some classes of the dynamic systems. Maltepe Üniversitesi. s. 150.#### Abstract

In this article the program control tracing algorithm is considered for some dynamic systems, which evolve on the
dynamic varieties. These algorithms are based on the differential equation systems construction, which have given in
advance the collection of the first integrals . The first algorithm is used the stochastic systems and the second is for the
determinate systems. The main idea of these algorithms is based on the first integral of SDE system definition, given by
prof. V.Doobko [1].
The program control of the stochastic system with the probability equaled to 1
Let us consider the SDE system with control:
dx(t) = [P (t; x(t)) + Q(t; x(t)) · s(t; x(t))] · dt + R(t; x(t)) · dw(t), (1)
where x(t) is a n-dimensional stochastic process, w(t) – is a m-dimensional standard Wiener process. The solution x(t) =
x(t; 0; xo, s) of the stochastic system (1) is called a program motion if it allows to stay on the given integrated variety
u(t; x(t; xo); ω) = u(0; xo) with the probability equaled to 1 for all time t at some s. This variety defines the first integrals
of the equations dx(t) = A(t, x(t))dt + B(t, x(t)) dw(t) with the given initial condition x(t; xo)
¯
¯
¯
t=0
= xo. Thus we shall
name the non-random function s = s(t; x(t)) as the program control for the dynamic stochastic system. The theory of the
first integral of SDE system in the prof. Doobko’s sense [1] allows to construct the new SDE system. In this system the
coefficients A and B are determined through the given dynamic variety surface for the system. This surface is invariant
for the system (1) with the probability equaled to 1, and it may be considered as the first integral collections of this SDE
system [2]. The congruence of the coefficients of the equations (1) and new equation make possible define the control
s(t; x(t)) = (s1, . . . , sn)
∗
and the reaction on random effect B(t; x(t)).
The continuous program control of the determinate system
As a rule definition of the program control of determinate systems is considered for the discreet points only, which
define the system position by the given periods of time. The specificity of our approach is that the controlled system is on
the given dynamic variability at any time.
We construct the class of the differential equations similar to [1]
dx(t) = A(t; x(t)) dt, which have the given first integrals collection n
u
l
(t; x)
oN
l=1
, N ≤ n. Then the program control
s(t; x(t)) for system
dx(t) = [P (t; x(t)) + Q(t; x(t)) · s(t; x(t))] dt
is as the solution of equation A(t; x(t)) = P (t; x(t))+Q(t; x(t))·s(t; x(t)). The conditions for the matrix Q and the invariant
surfaces are defined for the different dimensions control s.

#### Source

International Conference of Mathematical Sciences#### Collections

- Makale Koleksiyonu [586]

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