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{\bf RESTORATION OF CONTROLS IN NONLINEAR EVOLUTIONARY SYSTEMS UNDER
UNCERTAINTY\\[1.5ex] }
{\bf A.~I.~Korotkii\\}
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   The problem of dynamical restoration of unknown control $u\in U$ of a
nonlinear evolutionary system (1) is considered
$$
\dot x = f(t,x,u), \ u(t)\in P(t,x(t)),
\ t_0\leq t \leq \vartheta, \ x(t_0)=x_0. \eqno (1)
$$
The information for solving the problem consists of results of
approximate observations of moving sets $G(t)$, $x(t)\in G(t)$,
$t_0\leq t\leq\vartheta$. It is supposed that the exact state $x(t)$ of the
evolutionary system (1) is not measured and the observer is assumed not to
have enough information for estimation of the set $G(t)$ more exactly or find
appropriate statistical description for distribution of the state $x(t)$ in
the set $G(t)$. The problem belongs to the class of inverse problems of
dynamics [1] and is usually ill-posed [2,3].

   To solve the problem a dynamical positional regularizing algorithm
$D$ is suggested: $D(h,Z(t))=u_h(t)$, $t_0 \leq t \leq \vartheta$, $u_h\in U$,
(2) or (3) are realized as $h\to +0$
($Z[h,G(\cdot)]=\{Z(\cdot): \ae (Z(t),G(t)) \leq h,
\ t_0\leq t \leq \vartheta \} $)
$$
\sup \{\rho(u_h,u): Z(\cdot)\in Z[h,G(\cdot)] \} \to 0, \eqno (2)
$$
$$
\sup \{\nu(x_h(t),G(t)): Z(\cdot)\in Z[h,G(\cdot)],
\ t_0\leq t \leq \vartheta \} \to 0, \eqno (3)
$$
where $\rho$ is a criterion of control approximation (metric in space
$L^p$ on $[t_0,\vartheta]$, $1 \leq p < \infty$), $\ae$ is a criterion
of set approximation (Hausdorff metric),
$\nu$ is a criterion of state approximation (metric in the phase space),
$x_h(\cdot) = x_h(\cdot;t_0,x_0,u_h)$ is the
trajectory of the system (1) corresponding to the initial data $(t_0,x_0)$
($x_0 \in G(t_0)$) and control $u_h$. The algorithm is based on the methods
of the theory of positional control [4,1] and the theory of ill-posed
[2,3] problems.

Work supported by the Russian Foundation of Basic Researches (96-01-00846).
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REFERENCES
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1. Osipov Yu.S., Kryazhimskii A.V. Inverse problem of ordinary differential
   equations: dynamical solutions. London: Gordon and Breach. 1995.\\
2. Tikhonov A.N., Arsenin V.Ya. Solution of ill-posed problems. New York:
   Wiley. 1977. \\
3. Ivanov V.K., Vasin V.V., Tanana V.P. Theory of linear ill-posed problems
   and applications. Moscow: Nauka. 1978. \\
4. Krasovskii N.N., Subbotin A.I. Game-theoretical control problems.
   New York: Springer-Verlag. 1987.

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