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\title{Some Stable Algorithm for Solving Problems of Feedback Control and
Reconstruction for Distributed Parameter Systems}
\author{\it V.~I.~Maksimov}
\date{ }
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The problems of control and reconstruction of disturbances for systems
with distributed parameters are considered. The theory of these
problems has been much studied. The approach based on the method
of feedback control with a model developed by Ekaterinburg's school
is discussed. This approach is characterized by the following
two features: firstly, directedness to the work with nonlinear
dynamical systems and, secondly, orientation to construction of such solving
algorithms that are stable with respect to informational noises and
computational errors. These algorithms are required to be dynamical.
It means that they should work in ``real time'' and take into account
only previous history of the process.
Solving algorithms for two types of problems are constructed for
new classes of nonlinear systems according to the approach mentioned above.
Briefly, the essence of the problems under consideration
may be formulated by the following way. A motion of dynamical system
$\Sigma$ proceeds on a given time interval $T=[t_0,\vt]$. This system is
described by a parabolic inclusion. Its
trajectory $x(t)=x(t;u(\cd),v(\cd))$, $t \in T$
depends on a time-varying unknown input $v=v(t) \in Q \subset U_c$
and a control $u=u(t) \in P \subset U_c$.
$P$ and $Q$ are given sets from space of controls $U_c$. One takes a uniform
partition $\Delta = \{ \tau_i \}_{i=0}^m$ with diameter $\dl$.
Phase states $x(\tau_i)$ are measured
inaccurately at the moments $\tau_i$. The results of inaccurate measurements
$\xi_i$ satisfy inequalities $\chi(\xi_i,x(\tau_i)) \le h$, $i \in [0:m-1]$,
where $h$ is the value of level of informational noise,
$\chi(\cd,\cd)$ is the criterion for accuracy of measurements.
It is required to organize such process of control of system $\Sigma$
by feedback principle that allows to preserve given properties
of its trajectory under the action of any admissible input $v=v(\cd)$.
This is qualitative formulation of the control problems considered.
However, sometimes a system may be uncontrollable.
Its trajectory depends only on an unknown input $v=v(\cd)$.
In this case we consider the problem of reconstruction of the input.
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