An initial-boundary-value problem for a system of degenerate parabolic integro-differential equations is considered. The sufficient conditions for the existence and uniqueness of its generalized solution and for the existence of at least one optimal control for a given performance functional are obtained. A stable numerical solution to the initial-boundary-value problem is derived for a locally one-dimensional case and conditions are formulated for constructing a stable numerical algorithm of the optimal control problem on a class of piecewise-smooth control functions.

International audienceThe paper studies the problem of sliding mode control design for linear systems with incomplete and noisy measurements of the output and additive/multiplicative exogenous disturbances. First, we construct a linear minimax observer to have an estimate of the system's state with minimal worst-case error. Second, we establish the optimality of the constructed observer in the class of all observers represented by measurable functionals of the output. Finally, we propose an algorithm, generating continuous and discontinuous feedbacks, which steers the observer as close as possible to a given sliding hyperplane in finite time. The optimality (sub-optimality) of the designed feedbacks is proven for the case of bounded noises and additive (multiplicative) disturbances of L 2-class. The efficacy of the proposed algorithm is illustrated by a numerical example

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519.7 (075.8) An organizational two-level hierarchic system is used to examine control models for interregional migration under conditions of social risks. The Cauchy problem is used to solve the multidimensional transport equation (Problem A1) and a system of ordinary linear differential equations (Problem A2). For the models proposed, sufficient conditions for the existence of optimal control on the classes of functions designed are established, and stable numeral methods of searching for optimum solutions are developed.

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