A Guide to the Russian Academy of Sciences - University of Texas ...

Differential games.
For certain classes of game control systems with aftereffect, with distributed parameters
and with incomplete information, as well as abstract dynamical systems in Hilbert
spaces, the theorems of an alternative in apositional differential game have been
proved; they state solvability of one and only one of two control problems in a
differential convergence-evasion game; for the case of linear dynamics the regularity
conditions providing constructive solvability criteria for the problem of convergence
have been found; for linear systems with aftereffect, analogous conditions have
been stated for the problem of evasion (Iurii S. Osipov, A. V. Kryazhimskii, and
S. P. Okhezin). Several methods for designing solution control strategies with the
help of finite-dimensional approximation models have been proposed (V. I.
Maksimov, A. I. Korotkii, and D. A. Serkov). Some basic elements of the theory
of positional differential games (saddle point theorems for standard classes of
closed-loop strategies, stochastic approximation of mixed strategies, unification
theorems) have been developed for systems governed by ordinary differential
equations with non-lipschitzian right hand side (A. V. Kryazhimskii).
Inverse problems for control systems.
For systems governed by ordinary differential equations, several dynamical algorithms for
stable approximation of an input (control) on the basis of observation of states have
been described; in case all state coordinates are observed, the bounds for mean
square approximation within several classes of well-posedness have been obtained,
and their exact order with respect to the upper bound for observation noises has
been found; algorithms for stable approximation of a motion on the basis of
observations of a part of state coordinates have been constructed, their order
optimality has been shown, and conditions of asymptotical optimality have been
derived; relationships between different formulations of inverse problems and
solvability conditions are considered, nonsolvability examples have been presented
(Iurii S. Osipov and A. V. Kryazhimskii).
For control systems governed by linear and nonlinear parabolic equations with monotonous
operators, and parabolic variational inequalities, stable dynamical algorithms
providing approximations to distributed controls (included into a right hand side, in
general, nonlinearly) and coefficients, as well as to boundary ones have been
constructed; the cases where the deviations of observation results from a trajectory
are estimated in strong and weak metrics on the phase space have been considered;
stable dynamical solutions for the problem of tracking the location of sources have
been designed (Iurii S. Osipov, V. I. Maksimov, A. I. Korotkii, and A. V. Kim).
For systems governed by linear and nonlinear hyperbolic equations (special types of
nonlinearity) and variational inequalities, stable dynamical approximations to
distributed and boundary disturbances, and coefficients of the phase operator have
been constructed (Iurii S. Osipov and A. I Korotkii).
A general approach to the problems of stable dynamical approximation of time-dependent
inputs for systems with distributed parameters has been described (Iurii S. Osipov,
A. V. Kryazhimskii, and V. I. Maksimov).
Shape optimization.
The question of existence of an optimal (with respect to a certain functional) region for an
elliptic system and that of continuous dependence of a solution on a region have
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