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A problem of reconstruction of an unknown perturbation in a system of nonlinear ordinary differential equations is considered. The methods of solution of such problems are well known. In this paper we study a problem with two peculiarities. First, it is assumed that the phase coordinates of the dynamical system are measured (with error) at discrete sufficiently frequent times. Second, the only information known about the perturbation acting on the system is that its Euclidean norm is square integrable; i.e., the perturbation can be unbounded. Since the exact reconstruction is impossible under these assumptions, we design a solution algorithm that is stable under information noise and computation errors. The algorithm is based on the combination of elements of the theory of ill-posed problems with the extremal shift method known in the theory of positional differential games. © 2020 Krasovskii Institute of Mathematics and Mechanics. All rights reserved.

The article is devoted to the description of Academician Arkady Kryazhimskiy's life path. The facts of the scientific biography of Acad. Kryazhimskiy are presented with the emphasis on his outstanding contribution into the theory of dynamic inversion, the theory of differential games, and control theory. His personal talents in different spheres are also marked out.

The article considers dynamic identification of unknown characteristics in a third-order system. A realtime algorithm is proposed for the solution of the problem. The algorithm relies on constructs from stable dynamical inversion theory, which combines methods of the theory of ill-posed problems and positional control theory. In the proposed procedure, the reconstruction algorithm is represented as a control algorithm for some artificial dynamical system - a model. The model control is adapted to current observations so that its realization eventually "approximates" the unknown input. © 2013 Springer Science+Business Media New York.

Consideration was given to the problem of controlling a system of ordinary differential equations under incomplete information about the phase states. Given was an algorithm to solve it on the basis of a combination of the "real-time" reconstruction processes and feedback control. © 2013 Pleiades Publishing, Ltd.

We discuss a problem of the dynamic reconstruction of unmeasured coordinates of the phase vector and unknown controls in nonlinear vector equations with delay. A regularizing algorithm is proposed for the reconstruction of both controls and unmeasured coordinates simultaneously with the processes. The algorithm is stable with respect to information noises and computational errors. © 2013 Pleiades Publishing, Ltd.

We consider the problem of dynamic reconstruction of the input in a system described by a vector differential equation and nonlinear in the state variable. We indicate an algorithm that is stable under information noises and computational errors and is aimed at infinite system operation time. The algorithm is based on the dynamic regularization method. © 2013 Pleiades Publishing, Ltd.

The problem of reconstructing the intensity of a source function in a parabolic equation is considered. This problem is solved by an iterative algorithm based on a construction of feedback controls. The algorithm is robust with respect to information noise and computation errors. © Pleiades Publishing, Ltd., 2012.

The problem of reconstructing distributed inputs in linear parabolic equations is investigated. The algorithm proposed for solving this problem is stable with respect to informational disturbances and computational errors. It is based on the combination of methods from the theory of ill-posed problems and from the theory of positional control. The process of reconstructing unknown inputs implemented by the algorithm employs inaccurate measurements of phase coordinates of the system at discrete sufficiently frequent times. In the case when the input is a function of bounded variation, an upper estimate is established for the convergence rate. © 2012 Pleiades Publishing, Ltd.

For some classes of systems described by ordinary differential equations, a survey of algorithms for the dynamic reconstruction of inputs is presented. The algorithms described in the paper are stable with respect to information noises and computation errors; they are based on methods from the theory of ill-posed problems as well as on appropriate modifications of N. N. Krasovskii's principle of extremal aiming, which is known in the theory of guaranteed control. © 2011 Pleiades Publishing, Ltd.